%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP195^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:56:19 AM UTC 2026
% Result : Theorem 3.14s 1.64s
% Output : Refutation 3.14s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 4
% Syntax : Number of formulae : 28 ( 9 unt; 0 typ; 0 def)
% Number of atoms : 172 ( 68 equ; 0 cnn)
% Maximal formula atoms : 10 ( 6 avg)
% Number of connectives : 310 ( 39 ~; 22 |; 14 &; 219 @)
% ( 1 <=>; 14 =>; 0 <=; 1 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 167 ( 164 usr; 15 con; 0-3 aty)
% Number of variables : 0 ( 0 ^; 0 !; 0 ?; 0 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_6,type,
int: $tType ).
thf(type_def_7,type,
poly_int: $tType ).
thf(type_def_8,type,
poly_nat: $tType ).
thf(type_def_9,type,
poly_poly_int: $tType ).
thf(type_def_10,type,
poly_poly_nat: $tType ).
thf(type_def_11,type,
poly_poly_poly_real: $tType ).
thf(type_def_12,type,
poly_poly_real: $tType ).
thf(type_def_13,type,
poly_real: $tType ).
thf(type_def_14,type,
real: $tType ).
thf(type_def_16,type,
set_real: $tType ).
thf(func_def_0,type,
poly_poly_poly_real: $tType ).
thf(func_def_1,type,
poly_poly_real: $tType ).
thf(func_def_2,type,
poly_poly_nat: $tType ).
thf(func_def_3,type,
poly_poly_int: $tType ).
thf(func_def_4,type,
poly_real: $tType ).
thf(func_def_5,type,
poly_nat: $tType ).
thf(func_def_6,type,
poly_int: $tType ).
thf(func_def_7,type,
set_real: $tType ).
thf(func_def_8,type,
real: $tType ).
thf(func_def_9,type,
nat: $tType ).
thf(func_def_10,type,
int: $tType ).
thf(func_def_11,type,
one_one_nat: nat ).
thf(func_def_12,type,
zero_zero_int: int ).
thf(func_def_13,type,
zero_zero_nat: nat ).
thf(func_def_14,type,
zero_zero_poly_int: poly_int ).
thf(func_def_15,type,
zero_zero_poly_nat: poly_nat ).
thf(func_def_16,type,
zero_z1549157189ly_int: poly_poly_int ).
thf(func_def_17,type,
zero_z1059985641ly_nat: poly_poly_nat ).
thf(func_def_18,type,
zero_z935034829y_real: poly_poly_poly_real ).
thf(func_def_19,type,
zero_z1423781445y_real: poly_poly_real ).
thf(func_def_20,type,
zero_zero_poly_real: poly_real ).
thf(func_def_21,type,
zero_zero_real: real ).
thf(func_def_22,type,
suc: nat > nat ).
thf(func_def_23,type,
ord_less_int: int > int > $o ).
thf(func_def_24,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_25,type,
ord_less_poly_int: poly_int > poly_int > $o ).
thf(func_def_26,type,
ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
thf(func_def_27,type,
ord_less_poly_real: poly_real > poly_real > $o ).
thf(func_def_28,type,
ord_less_real: real > real > $o ).
thf(func_def_29,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_30,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_31,type,
ord_less_eq_poly_int: poly_int > poly_int > $o ).
thf(func_def_32,type,
ord_le893774876y_real: poly_poly_real > poly_poly_real > $o ).
thf(func_def_33,type,
ord_le1180086932y_real: poly_real > poly_real > $o ).
thf(func_def_34,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_35,type,
is_zero_int: poly_int > $o ).
thf(func_def_36,type,
is_zero_nat: poly_nat > $o ).
thf(func_def_37,type,
is_zero_poly_real: poly_poly_real > $o ).
thf(func_def_38,type,
is_zero_real: poly_real > $o ).
thf(func_def_39,type,
order_int: int > poly_int > nat ).
thf(func_def_40,type,
order_poly_int: poly_int > poly_poly_int > nat ).
thf(func_def_41,type,
order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
thf(func_def_42,type,
order_poly_real: poly_real > poly_poly_real > nat ).
thf(func_def_43,type,
order_real: real > poly_real > nat ).
thf(func_def_44,type,
pderiv_int: poly_int > poly_int ).
thf(func_def_45,type,
pderiv_nat: poly_nat > poly_nat ).
thf(func_def_46,type,
pderiv_poly_real: poly_poly_real > poly_poly_real ).
thf(func_def_47,type,
pderiv_real: poly_real > poly_real ).
thf(func_def_48,type,
poly_int2: poly_int > int > int ).
thf(func_def_49,type,
poly_nat2: poly_nat > nat > nat ).
thf(func_def_50,type,
poly_poly_int2: poly_poly_int > poly_int > poly_int ).
thf(func_def_51,type,
poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
thf(func_def_52,type,
poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
thf(func_def_53,type,
poly_poly_real2: poly_poly_real > poly_real > poly_real ).
thf(func_def_54,type,
poly_real2: poly_real > real > real ).
thf(func_def_55,type,
poly_cutoff_int: nat > poly_int > poly_int ).
thf(func_def_56,type,
poly_cutoff_nat: nat > poly_nat > poly_nat ).
thf(func_def_57,type,
poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
thf(func_def_58,type,
poly_cutoff_real: nat > poly_real > poly_real ).
thf(func_def_59,type,
poly_shift_int: nat > poly_int > poly_int ).
thf(func_def_60,type,
poly_shift_nat: nat > poly_nat > poly_nat ).
thf(func_def_61,type,
poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
thf(func_def_62,type,
poly_shift_real: nat > poly_real > poly_real ).
thf(func_def_63,type,
reflect_poly_int: poly_int > poly_int ).
thf(func_def_64,type,
reflect_poly_nat: poly_nat > poly_nat ).
thf(func_def_65,type,
reflec943087998ly_int: poly_poly_int > poly_poly_int ).
thf(func_def_66,type,
reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
thf(func_def_67,type,
reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
thf(func_def_68,type,
reflec1522834046y_real: poly_poly_real > poly_poly_real ).
thf(func_def_69,type,
reflect_poly_real: poly_real > poly_real ).
thf(func_def_70,type,
rsquarefree_real: poly_real > $o ).
thf(func_def_71,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_72,type,
sturm_1953858694_cross: poly_real > real > real > int ).
thf(func_def_73,type,
sturm_469514713_r_pos: poly_real > real > $o ).
thf(func_def_74,type,
sturm_1279781401iation: real > real > int ).
thf(func_def_75,type,
member_real: real > set_real > $o ).
thf(func_def_76,type,
p: poly_real ).
thf(func_def_77,type,
x: real ).
thf(func_def_92,type,
sP0: real > real > real > $o ).
thf(func_def_93,type,
sP1: real > real > real > $o ).
thf(func_def_94,type,
sP2: real > real > real > $o ).
thf(func_def_95,type,
sK3: ( nat > nat ) > nat ).
thf(func_def_96,type,
sK4: ( nat > nat ) > nat ).
thf(func_def_97,type,
sK5: real > real ).
thf(func_def_98,type,
sK6: ( nat > real ) > nat ).
thf(func_def_99,type,
sK7: ( nat > real ) > nat ).
thf(func_def_100,type,
sK8: ( nat > real ) > nat ).
thf(func_def_101,type,
sK9: ( nat > real ) > nat ).
thf(func_def_102,type,
sK10: ( real > real ) > real ).
thf(func_def_103,type,
sK11: ( real > real ) > real ).
thf(func_def_104,type,
sK12: ( nat > real ) > nat ).
thf(func_def_105,type,
sK13: ( nat > real ) > nat ).
thf(func_def_106,type,
sK14: real > real ).
thf(func_def_107,type,
sK15: real > poly_real > real ).
thf(func_def_108,type,
sK16: ( real > real ) > real ).
thf(func_def_109,type,
sK17: ( real > real ) > real ).
thf(func_def_110,type,
sK18: ( real > real > $o ) > real ).
thf(func_def_111,type,
sK19: ( real > real > $o ) > real ).
thf(func_def_112,type,
sK20: ( real > real > $o ) > real ).
thf(func_def_113,type,
sK21: ( real > real > $o ) > real ).
thf(func_def_114,type,
sK22: ( real > real > $o ) > real ).
thf(func_def_115,type,
sK23: ( nat > nat ) > nat ).
thf(func_def_116,type,
sK24: ( nat > nat ) > nat ).
thf(func_def_117,type,
sK25: ( real > nat ) > real ).
thf(func_def_118,type,
sK26: ( real > nat ) > real ).
thf(func_def_119,type,
sK27: real > real > real ).
thf(func_def_120,type,
sK28: poly_real > real > real > real ).
thf(func_def_121,type,
sK29: ( real > real ) > real ).
thf(func_def_122,type,
sK30: ( real > real ) > real ).
thf(func_def_123,type,
sK31: ( nat > real ) > nat ).
thf(func_def_124,type,
sK32: ( nat > real ) > nat ).
thf(func_def_125,type,
sK33: ( real > nat ) > real ).
thf(func_def_126,type,
sK34: ( real > nat ) > real ).
thf(func_def_127,type,
sK35: real > real ).
thf(func_def_128,type,
sK36: ( real > nat ) > real ).
thf(func_def_129,type,
sK37: ( real > nat ) > real ).
thf(func_def_130,type,
sK38: ( nat > real ) > nat ).
thf(func_def_131,type,
sK39: ( nat > real ) > nat ).
thf(func_def_132,type,
sK40: nat > nat ).
thf(func_def_133,type,
sK41: ( nat > real ) > nat ).
thf(func_def_134,type,
sK42: ( nat > real ) > nat ).
thf(func_def_135,type,
sK43: ( real > real > $o ) > real ).
thf(func_def_136,type,
sK44: ( real > real > $o ) > real ).
thf(func_def_137,type,
sK45: ( real > real > $o ) > real ).
thf(func_def_138,type,
sK46: ( real > real > $o ) > real ).
thf(func_def_139,type,
sK47: ( real > nat ) > real ).
thf(func_def_140,type,
sK48: ( real > nat ) > real ).
thf(func_def_141,type,
sK49: ( real > nat ) > real ).
thf(func_def_142,type,
sK50: ( real > nat ) > real ).
thf(func_def_143,type,
sK51: ( real > nat ) > real ).
thf(func_def_144,type,
sK52: ( real > nat ) > real ).
thf(func_def_145,type,
sK53: ( nat > $o ) > nat ).
thf(func_def_146,type,
sK54: real > real ).
thf(func_def_147,type,
sK55: real > real ).
thf(func_def_148,type,
sK56: poly_real > real ).
thf(func_def_149,type,
sK57: ( nat > $o ) > nat ).
thf(func_def_150,type,
sK58: poly_real > real > real > real ).
thf(func_def_151,type,
sK59: ( real > nat ) > real ).
thf(func_def_152,type,
sK60: ( real > nat ) > real ).
thf(func_def_153,type,
sK61: poly_real > real > real > real ).
thf(func_def_154,type,
sK62: poly_real > real > real > real ).
thf(func_def_155,type,
sK63: poly_real > real > real > real ).
thf(func_def_156,type,
sK64: ( nat > nat ) > nat ).
thf(func_def_157,type,
sK65: ( nat > nat ) > nat ).
thf(func_def_158,type,
sK66: poly_real > real > real ).
thf(func_def_159,type,
sK67: real > real ).
thf(func_def_160,type,
sK68: ( real > nat ) > real ).
thf(func_def_161,type,
sK69: ( real > nat ) > real ).
thf(func_def_162,type,
sK70: ( real > real ) > real ).
thf(func_def_163,type,
sK71: ( real > real ) > real ).
thf(func_def_164,type,
sK72: ( nat > real ) > nat ).
thf(func_def_165,type,
sK73: ( nat > real ) > nat ).
thf(func_def_166,type,
sK74: real > real > real ).
thf(func_def_167,type,
sK75: ( real > real ) > real ).
thf(func_def_168,type,
sK76: ( real > real ) > real ).
thf(func_def_169,type,
sK77: ( nat > nat > $o ) > nat ).
thf(func_def_170,type,
sK78: ( nat > nat > $o ) > nat ).
thf(func_def_171,type,
sK79: ( nat > nat > $o ) > nat ).
thf(func_def_172,type,
sK80: ( nat > nat > $o ) > nat ).
thf(func_def_173,type,
sK81: ( nat > nat > $o ) > nat ).
thf(func_def_174,type,
sK82: ( real > real ) > real ).
thf(func_def_175,type,
sK83: ( real > real ) > real ).
thf(func_def_176,type,
sK84: ( nat > nat ) > nat ).
thf(func_def_177,type,
sK85: ( nat > nat ) > nat ).
thf(func_def_178,type,
sK86: ( nat > real ) > nat ).
thf(func_def_179,type,
sK87: ( nat > real ) > nat ).
thf(func_def_180,type,
sK88: ( real > real ) > real ).
thf(func_def_181,type,
sK89: ( real > real ) > real ).
thf(func_def_182,type,
sK90: ( nat > $o ) > nat ).
thf(func_def_183,type,
sK91: poly_real > real > real > real ).
thf(func_def_184,type,
sK92: poly_real > real > real > real ).
thf(func_def_185,type,
sK93: poly_real > real > real > real ).
thf(func_def_186,type,
sK94: ( real > real ) > real ).
thf(func_def_187,type,
sK95: ( real > real ) > real ).
thf(func_def_189,type,
ph97:
!>[X0: $tType] : X0 ).
thf(f1651,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1632,f1618]) ).
thf(f1618,plain,
( ( sturm_469514713_r_pos @ p @ x )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f1617,f1616]) ).
thf(f1616,plain,
( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true ),
inference(subsumption_resolution,[status(thm)],[f1615,f1530]) ).
thf(f1530,plain,
( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true )
| ( ( sturm_469514713_r_pos @ p @ x )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1072]) ).
thf(f1072,plain,
( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true )
| ( ( sturm_469514713_r_pos @ p @ x )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f953]) ).
thf(f953,plain,
( ( ( sturm_469514713_r_pos @ p @ x )
= $true )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f952]) ).
thf(f952,plain,
( ( sturm_469514713_r_pos @ p @ x )
=> ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( sturm_469514713_r_pos @ p @ x )
=> ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.JHQKkcEisk/DTF2THF_31714.p',fact_1__092_060open_062sign__r__pos_Ap_Ax_A_092_060Longrightarrow_062_A0_A_060_Apoly_Ap_Ax_092_060close_062) ).
thf(f1615,plain,
( ( ( sturm_469514713_r_pos @ p @ x )
= $true )
| ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true ) ),
inference(subsumption_resolution,[status(thm)],[f1353,f1510]) ).
thf(f1510,plain,
( ( poly_real2 @ p @ x )
!= zero_zero_real ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( poly_real2 @ p @ x )
!= zero_zero_real ),
file('/export/starexec/sandbox2/tmp/tmp.JHQKkcEisk/DTF2THF_31714.p',fact_0_False) ).
thf(f1353,plain,
( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true )
| ( ( sturm_469514713_r_pos @ p @ x )
= $true )
| ( ( poly_real2 @ p @ x )
= zero_zero_real ) ),
inference(cnf_transformation,[status(thm)],[f1191]) ).
thf(f1191,plain,
( ( ( ( ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x )
!= $true )
& ( ( poly_real2 @ p @ x )
= zero_zero_real ) )
| ( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
!= $true )
& ( ( poly_real2 @ p @ x )
!= zero_zero_real ) )
| ( ( sturm_469514713_r_pos @ p @ x )
!= $true ) )
& ( ( ( sturm_469514713_r_pos @ p @ x )
= $true )
| ( ( ( ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x )
= $true )
| ( ( poly_real2 @ p @ x )
!= zero_zero_real ) )
& ( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true )
| ( ( poly_real2 @ p @ x )
= zero_zero_real ) ) ) ) ),
inference(flattening,[status(thm)],[f1190]) ).
thf(f1190,plain,
( ( ( ( ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x )
!= $true )
& ( ( poly_real2 @ p @ x )
= zero_zero_real ) )
| ( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
!= $true )
& ( ( poly_real2 @ p @ x )
!= zero_zero_real ) )
| ( ( sturm_469514713_r_pos @ p @ x )
!= $true ) )
& ( ( ( sturm_469514713_r_pos @ p @ x )
= $true )
| ( ( ( ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x )
= $true )
| ( ( poly_real2 @ p @ x )
!= zero_zero_real ) )
& ( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true )
| ( ( poly_real2 @ p @ x )
= zero_zero_real ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1157]) ).
thf(f1157,plain,
( ( ( ( ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x )
!= $true )
& ( ( poly_real2 @ p @ x )
= zero_zero_real ) )
| ( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
!= $true )
& ( ( poly_real2 @ p @ x )
!= zero_zero_real ) ) )
<=> ( ( sturm_469514713_r_pos @ p @ x )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f625]) ).
thf(f625,plain,
~ ( ~ ( ( ( ( poly_real2 @ p @ x )
= zero_zero_real )
=> ( ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x )
= $true ) )
& ( ( ( poly_real2 @ p @ x )
!= zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true ) ) )
<~> ( ( sturm_469514713_r_pos @ p @ x )
= $true ) ),
inference(fool_elimination,[status(thm)],[f624]) ).
thf(f624,plain,
~ ( ( ( sturm_469514713_r_pos @ p @ x )
!= ( ~ ( ( ( ( poly_real2 @ p @ x )
!= zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) ) )
& ( ( ( poly_real2 @ p @ x )
= zero_zero_real )
=> ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x ) ) ) ) ) ),
inference(rectify,[status(thm)],[f347]) ).
thf(f347,negated_conjecture,
~ ( ( ( sturm_469514713_r_pos @ p @ x )
!= ( ~ ( ( ( ( poly_real2 @ p @ x )
!= zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) ) )
& ( ( ( poly_real2 @ p @ x )
= zero_zero_real )
=> ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f346]) ).
thf(f346,conjecture,
( ( sturm_469514713_r_pos @ p @ x )
!= ( ~ ( ( ( ( poly_real2 @ p @ x )
!= zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) ) )
& ( ( ( poly_real2 @ p @ x )
= zero_zero_real )
=> ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.JHQKkcEisk/DTF2THF_31714.p',conj_0) ).
thf(f1617,plain,
( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
!= $true )
| ( ( sturm_469514713_r_pos @ p @ x )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f1356,f1510]) ).
thf(f1356,plain,
( ( ( sturm_469514713_r_pos @ p @ x )
!= $true )
| ( ( poly_real2 @ p @ x )
= zero_zero_real )
| ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1191]) ).
thf(f1632,plain,
( ( sturm_469514713_r_pos @ p @ x )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1619]) ).
thf(f1619,plain,
( ( ( sturm_469514713_r_pos @ p @ x )
= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1342,f1616]) ).
thf(f1342,plain,
( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
!= $true )
| ( ( sturm_469514713_r_pos @ p @ x )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1037]) ).
thf(f1037,plain,
( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
!= $true )
| ( ( sturm_469514713_r_pos @ p @ x )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f464]) ).
thf(f464,plain,
( ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
= $true )
=> ( ( sturm_469514713_r_pos @ p @ x )
= $true ) ),
inference(fool_elimination,[status(thm)],[f463]) ).
thf(f463,plain,
( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
=> ( sturm_469514713_r_pos @ p @ x ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
=> ( sturm_469514713_r_pos @ p @ x ) ),
file('/export/starexec/sandbox2/tmp/tmp.JHQKkcEisk/DTF2THF_31714.p',fact_2__092_060open_0620_A_060_Apoly_Ap_Ax_A_092_060Longrightarrow_062_Asign__r__pos_Ap_Ax_092_060close_062) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP195^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n010.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Tue Feb 24 08:29:18 EST 2026
% 0.14/0.33 % CPUTime :
% 0.14/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.44 ---- Original DTF file ---
% 0.18/0.44 thf(spec,logic,$$dhol).
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % File : ITP195^1 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.44 % Domain : Interactive Theorem Proving
% 0.18/0.44 % Problem : Sledgehammer Sturm_Tarski problem prob_450__5874512_1
% 0.18/0.44 % Version : Especial.
% 0.18/0.44 % English :
% 0.18/0.44
% 0.18/0.44 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.44 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.44 % Source : [Des21]
% 0.18/0.44 % Names : Sturm_Tarski/prob_450__5874512_1 [Des21]
% 0.18/0.44
% 0.18/0.44 % Status : Theorem
% 0.18/0.44 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.18/0.44 % Syntax : Number of formulae : 424 ( 156 unt; 78 typ; 0 def)
% 0.18/0.44 % Number of atoms : 1005 ( 321 equ; 0 cnn)
% 0.18/0.44 % Maximal formula atoms : 12 ( 2 avg)
% 0.18/0.44 % Number of connectives : 2572 ( 149 ~; 32 |; 37 &;1859 @)
% 0.18/0.44 % ( 0 <=>; 495 =>; 0 <=; 0 <~>)
% 0.18/0.44 % Maximal formula depth : 14 ( 7 avg)
% 0.18/0.44 % Number of types : 12 ( 11 usr)
% 0.18/0.44 % Number of type conns : 168 ( 168 >; 0 *; 0 +; 0 <<)
% 0.18/0.44 % Number of symbols : 68 ( 67 usr; 13 con; 0-3 aty)
% 0.18/0.44 % Number of variables : 923 ( 53 ^; 841 !; 29 ?; 923 :)
% 0.18/0.44 % SPC : TH0_THM_EQU_NAR
% 0.18/0.44
% 0.18/0.44 % Comments : This file was generated by Sledgehammer 2021-02-23 15:42:23.162
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % Could-be-implicit typings (11)
% 0.18/0.44 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 0.18/0.44 poly_poly_poly_real: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.18/0.44 poly_poly_real: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 0.18/0.44 poly_poly_nat: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
% 0.18/0.44 poly_poly_int: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.44 poly_real: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.18/0.44 poly_nat: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.18/0.44 poly_int: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.18/0.44 set_real: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Real__Oreal,type,
% 0.18/0.44 real: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Nat__Onat,type,
% 0.18/0.44 nat: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Int__Oint,type,
% 0.18/0.44 int: $tType ).
% 0.18/0.44
% 0.18/0.44 % Explicit typings (67)
% 0.18/0.44 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.18/0.44 one_one_nat: nat ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.18/0.44 zero_zero_int: int ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.18/0.44 zero_zero_nat: nat ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.18/0.44 zero_zero_poly_int: poly_int ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.18/0.44 zero_zero_poly_nat: poly_nat ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
% 0.18/0.44 zero_z1549157189ly_int: poly_poly_int ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 0.18/0.44 zero_z1059985641ly_nat: poly_poly_nat ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 0.18/0.44 zero_z935034829y_real: poly_poly_poly_real ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.18/0.44 zero_z1423781445y_real: poly_poly_real ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.44 zero_zero_poly_real: poly_real ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.18/0.44 zero_zero_real: real ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Nat_OSuc,type,
% 0.18/0.44 suc: nat > nat ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.18/0.44 ord_less_int: int > int > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.18/0.44 ord_less_nat: nat > nat > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.18/0.44 ord_less_poly_int: poly_int > poly_int > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.18/0.44 ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.44 ord_less_poly_real: poly_real > poly_real > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.18/0.44 ord_less_real: real > real > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.18/0.44 ord_less_eq_int: int > int > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.18/0.44 ord_less_eq_nat: nat > nat > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.18/0.44 ord_less_eq_poly_int: poly_int > poly_int > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.18/0.44 ord_le893774876y_real: poly_poly_real > poly_poly_real > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.44 ord_le1180086932y_real: poly_real > poly_real > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.18/0.44 ord_less_eq_real: real > real > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Ois__zero_001t__Int__Oint,type,
% 0.18/0.44 is_zero_int: poly_int > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat,type,
% 0.18/0.44 is_zero_nat: poly_nat > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.44 is_zero_poly_real: poly_poly_real > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Ois__zero_001t__Real__Oreal,type,
% 0.18/0.44 is_zero_real: poly_real > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Oorder_001t__Int__Oint,type,
% 0.18/0.44 order_int: int > poly_int > nat ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.18/0.44 order_poly_int: poly_int > poly_poly_int > nat ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.18/0.44 order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.45 order_poly_real: poly_real > poly_poly_real > nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Oorder_001t__Real__Oreal,type,
% 0.18/0.45 order_real: real > poly_real > nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opderiv_001t__Int__Oint,type,
% 0.18/0.45 pderiv_int: poly_int > poly_int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opderiv_001t__Nat__Onat,type,
% 0.18/0.45 pderiv_nat: poly_nat > poly_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opderiv_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.45 pderiv_poly_real: poly_poly_real > poly_poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opderiv_001t__Real__Oreal,type,
% 0.18/0.45 pderiv_real: poly_real > poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly_001t__Int__Oint,type,
% 0.18/0.45 poly_int2: poly_int > int > int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
% 0.18/0.45 poly_nat2: poly_nat > nat > nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.18/0.45 poly_poly_int2: poly_poly_int > poly_int > poly_int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.18/0.45 poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.18/0.45 poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.45 poly_poly_real2: poly_poly_real > poly_real > poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
% 0.18/0.45 poly_real2: poly_real > real > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly__cutoff_001t__Int__Oint,type,
% 0.18/0.45 poly_cutoff_int: nat > poly_int > poly_int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat,type,
% 0.18/0.45 poly_cutoff_nat: nat > poly_nat > poly_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.45 poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly__cutoff_001t__Real__Oreal,type,
% 0.18/0.45 poly_cutoff_real: nat > poly_real > poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly__shift_001t__Int__Oint,type,
% 0.18/0.45 poly_shift_int: nat > poly_int > poly_int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat,type,
% 0.18/0.45 poly_shift_nat: nat > poly_nat > poly_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.45 poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Opoly__shift_001t__Real__Oreal,type,
% 0.18/0.45 poly_shift_real: nat > poly_real > poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Oreflect__poly_001t__Int__Oint,type,
% 0.18/0.45 reflect_poly_int: poly_int > poly_int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
% 0.18/0.45 reflect_poly_nat: poly_nat > poly_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.18/0.45 reflec943087998ly_int: poly_poly_int > poly_poly_int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.18/0.45 reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.18/0.45 reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.18/0.45 reflec1522834046y_real: poly_poly_real > poly_poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal,type,
% 0.18/0.45 reflect_poly_real: poly_real > poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Polynomial_Orsquarefree_001t__Real__Oreal,type,
% 0.18/0.45 rsquarefree_real: poly_real > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.18/0.45 collect_real: ( real > $o ) > set_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Ocross,type,
% 0.18/0.45 sturm_1953858694_cross: poly_real > real > real > int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Osign__r__pos,type,
% 0.18/0.45 sturm_469514713_r_pos: poly_real > real > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Ovariation,type,
% 0.18/0.45 sturm_1279781401iation: real > real > int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Real__Oreal,type,
% 0.18/0.45 member_real: real > set_real > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_p,type,
% 0.18/0.45 p: poly_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_x,type,
% 0.18/0.45 x: real ).
% 0.18/0.45
% 0.18/0.45 % Relevant facts (345)
% 0.18/0.45 thf(fact_0_False,axiom,
% 0.18/0.45 ( ( poly_real2 @ p @ x )
% 0.18/0.45 != zero_zero_real ) ).
% 0.18/0.45
% 0.18/0.45 % False
% 0.18/0.45 thf(fact_1__092_060open_062sign__r__pos_Ap_Ax_A_092_060Longrightarrow_062_A0_A_060_Apoly_Ap_Ax_092_060close_062,axiom,
% 0.18/0.45 ( ( sturm_469514713_r_pos @ p @ x )
% 0.18/0.45 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % \<open>sign_r_pos p x \<Longrightarrow> 0 < poly p x\<close>
% 0.18/0.45 thf(fact_2__092_060open_0620_A_060_Apoly_Ap_Ax_A_092_060Longrightarrow_062_Asign__r__pos_Ap_Ax_092_060close_062,axiom,
% 0.18/0.45 ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
% 0.18/0.45 => ( sturm_469514713_r_pos @ p @ x ) ) ).
% 0.18/0.45
% 0.18/0.45 % \<open>0 < poly p x \<Longrightarrow> sign_r_pos p x\<close>
% 0.18/0.45 thf(fact_3_assms,axiom,
% 0.18/0.45 p != zero_zero_poly_real ).
% 0.18/0.45
% 0.18/0.45 % assms
% 0.18/0.45 thf(fact_4_poly__0,axiom,
% 0.18/0.45 ! [X: poly_nat] :
% 0.18/0.45 ( ( poly_poly_nat2 @ zero_z1059985641ly_nat @ X )
% 0.18/0.45 = zero_zero_poly_nat ) ).
% 0.18/0.45
% 0.18/0.45 % poly_0
% 0.18/0.45 thf(fact_5_poly__0,axiom,
% 0.18/0.45 ! [X: poly_int] :
% 0.18/0.45 ( ( poly_poly_int2 @ zero_z1549157189ly_int @ X )
% 0.18/0.45 = zero_zero_poly_int ) ).
% 0.18/0.45
% 0.18/0.45 % poly_0
% 0.18/0.45 thf(fact_6_poly__0,axiom,
% 0.18/0.45 ! [X: poly_poly_real] :
% 0.18/0.45 ( ( poly_poly_poly_real2 @ zero_z935034829y_real @ X )
% 0.18/0.45 = zero_z1423781445y_real ) ).
% 0.18/0.45
% 0.18/0.45 % poly_0
% 0.18/0.45 thf(fact_7_poly__0,axiom,
% 0.18/0.45 ! [X: poly_real] :
% 0.18/0.45 ( ( poly_poly_real2 @ zero_z1423781445y_real @ X )
% 0.18/0.45 = zero_zero_poly_real ) ).
% 0.18/0.45
% 0.18/0.45 % poly_0
% 0.18/0.45 thf(fact_8_poly__0,axiom,
% 0.18/0.45 ! [X: int] :
% 0.18/0.45 ( ( poly_int2 @ zero_zero_poly_int @ X )
% 0.18/0.45 = zero_zero_int ) ).
% 0.18/0.45
% 0.18/0.45 % poly_0
% 0.18/0.45 thf(fact_9_poly__0,axiom,
% 0.18/0.45 ! [X: nat] :
% 0.18/0.45 ( ( poly_nat2 @ zero_zero_poly_nat @ X )
% 0.18/0.45 = zero_zero_nat ) ).
% 0.18/0.45
% 0.18/0.45 % poly_0
% 0.18/0.45 thf(fact_10_poly__0,axiom,
% 0.18/0.45 ! [X: real] :
% 0.18/0.45 ( ( poly_real2 @ zero_zero_poly_real @ X )
% 0.18/0.45 = zero_zero_real ) ).
% 0.18/0.45
% 0.18/0.45 % poly_0
% 0.18/0.45 thf(fact_11_not__gr__zero,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.18/0.45 = ( N = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % not_gr_zero
% 0.18/0.45 thf(fact_12_poly__IVT__neg,axiom,
% 0.18/0.45 ! [A: real,B: real,P: poly_real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A ) )
% 0.18/0.45 => ( ( ord_less_real @ ( poly_real2 @ P @ B ) @ zero_zero_real )
% 0.18/0.45 => ? [X2: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ X2 )
% 0.18/0.45 & ( ord_less_real @ X2 @ B )
% 0.18/0.45 & ( ( poly_real2 @ P @ X2 )
% 0.18/0.45 = zero_zero_real ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_IVT_neg
% 0.18/0.45 thf(fact_13_poly__IVT__pos,axiom,
% 0.18/0.45 ! [A: real,B: real,P: poly_real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_real @ ( poly_real2 @ P @ A ) @ zero_zero_real )
% 0.18/0.45 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B ) )
% 0.18/0.45 => ? [X2: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ X2 )
% 0.18/0.45 & ( ord_less_real @ X2 @ B )
% 0.18/0.45 & ( ( poly_real2 @ P @ X2 )
% 0.18/0.45 = zero_zero_real ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_IVT_pos
% 0.18/0.45 thf(fact_14_poly__all__0__iff__0,axiom,
% 0.18/0.45 ! [P: poly_poly_int] :
% 0.18/0.45 ( ( ! [X3: poly_int] :
% 0.18/0.45 ( ( poly_poly_int2 @ P @ X3 )
% 0.18/0.45 = zero_zero_poly_int ) )
% 0.18/0.45 = ( P = zero_z1549157189ly_int ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_all_0_iff_0
% 0.18/0.45 thf(fact_15_poly__all__0__iff__0,axiom,
% 0.18/0.45 ! [P: poly_poly_poly_real] :
% 0.18/0.45 ( ( ! [X3: poly_poly_real] :
% 0.18/0.45 ( ( poly_poly_poly_real2 @ P @ X3 )
% 0.18/0.45 = zero_z1423781445y_real ) )
% 0.18/0.45 = ( P = zero_z935034829y_real ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_all_0_iff_0
% 0.18/0.45 thf(fact_16_poly__all__0__iff__0,axiom,
% 0.18/0.45 ! [P: poly_real] :
% 0.18/0.45 ( ( ! [X3: real] :
% 0.18/0.45 ( ( poly_real2 @ P @ X3 )
% 0.18/0.45 = zero_zero_real ) )
% 0.18/0.45 = ( P = zero_zero_poly_real ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_all_0_iff_0
% 0.18/0.45 thf(fact_17_poly__all__0__iff__0,axiom,
% 0.18/0.45 ! [P: poly_poly_real] :
% 0.18/0.45 ( ( ! [X3: poly_real] :
% 0.18/0.45 ( ( poly_poly_real2 @ P @ X3 )
% 0.18/0.45 = zero_zero_poly_real ) )
% 0.18/0.45 = ( P = zero_z1423781445y_real ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_all_0_iff_0
% 0.18/0.45 thf(fact_18_poly__all__0__iff__0,axiom,
% 0.18/0.45 ! [P: poly_int] :
% 0.18/0.45 ( ( ! [X3: int] :
% 0.18/0.45 ( ( poly_int2 @ P @ X3 )
% 0.18/0.45 = zero_zero_int ) )
% 0.18/0.45 = ( P = zero_zero_poly_int ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_all_0_iff_0
% 0.18/0.45 thf(fact_19_less__numeral__extra_I3_J,axiom,
% 0.18/0.45 ~ ( ord_less_poly_real @ zero_zero_poly_real @ zero_zero_poly_real ) ).
% 0.18/0.45
% 0.18/0.45 % less_numeral_extra(3)
% 0.18/0.45 thf(fact_20_less__numeral__extra_I3_J,axiom,
% 0.18/0.45 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.18/0.45
% 0.18/0.45 % less_numeral_extra(3)
% 0.18/0.45 thf(fact_21_less__numeral__extra_I3_J,axiom,
% 0.18/0.45 ~ ( ord_less_poly_int @ zero_zero_poly_int @ zero_zero_poly_int ) ).
% 0.18/0.45
% 0.18/0.45 % less_numeral_extra(3)
% 0.18/0.45 thf(fact_22_less__numeral__extra_I3_J,axiom,
% 0.18/0.45 ~ ( ord_le38482960y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ) ).
% 0.18/0.45
% 0.18/0.45 % less_numeral_extra(3)
% 0.18/0.45 thf(fact_23_less__numeral__extra_I3_J,axiom,
% 0.18/0.45 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.18/0.45
% 0.18/0.45 % less_numeral_extra(3)
% 0.18/0.45 thf(fact_24_less__numeral__extra_I3_J,axiom,
% 0.18/0.45 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.18/0.45
% 0.18/0.45 % less_numeral_extra(3)
% 0.18/0.45 thf(fact_25_field__lbound__gt__zero,axiom,
% 0.18/0.45 ! [D1: real,D2: real] :
% 0.18/0.45 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.18/0.45 => ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.18/0.45 => ? [E: real] :
% 0.18/0.45 ( ( ord_less_real @ zero_zero_real @ E )
% 0.18/0.45 & ( ord_less_real @ E @ D1 )
% 0.18/0.45 & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % field_lbound_gt_zero
% 0.18/0.45 thf(fact_26_gr__zeroI,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( N != zero_zero_nat )
% 0.18/0.45 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % gr_zeroI
% 0.18/0.45 thf(fact_27_not__less__zero,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.18/0.45
% 0.18/0.45 % not_less_zero
% 0.18/0.45 thf(fact_28_gr__implies__not__zero,axiom,
% 0.18/0.45 ! [M: nat,N: nat] :
% 0.18/0.45 ( ( ord_less_nat @ M @ N )
% 0.18/0.45 => ( N != zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % gr_implies_not_zero
% 0.18/0.45 thf(fact_29_zero__less__iff__neq__zero,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.45 = ( N != zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % zero_less_iff_neq_zero
% 0.18/0.45 thf(fact_30_pderiv__0,axiom,
% 0.18/0.45 ( ( pderiv_real @ zero_zero_poly_real )
% 0.18/0.45 = zero_zero_poly_real ) ).
% 0.18/0.45
% 0.18/0.45 % pderiv_0
% 0.18/0.45 thf(fact_31_pderiv__0,axiom,
% 0.18/0.45 ( ( pderiv_nat @ zero_zero_poly_nat )
% 0.18/0.45 = zero_zero_poly_nat ) ).
% 0.18/0.45
% 0.18/0.45 % pderiv_0
% 0.18/0.45 thf(fact_32_pderiv__0,axiom,
% 0.18/0.45 ( ( pderiv_int @ zero_zero_poly_int )
% 0.18/0.45 = zero_zero_poly_int ) ).
% 0.18/0.45
% 0.18/0.45 % pderiv_0
% 0.18/0.45 thf(fact_33_pderiv__0,axiom,
% 0.18/0.45 ( ( pderiv_poly_real @ zero_z1423781445y_real )
% 0.18/0.45 = zero_z1423781445y_real ) ).
% 0.18/0.45
% 0.18/0.45 % pderiv_0
% 0.18/0.45 thf(fact_34_zero__reorient,axiom,
% 0.18/0.45 ! [X: real] :
% 0.18/0.45 ( ( zero_zero_real = X )
% 0.18/0.45 = ( X = zero_zero_real ) ) ).
% 0.18/0.45
% 0.18/0.45 % zero_reorient
% 0.18/0.45 thf(fact_35_zero__reorient,axiom,
% 0.18/0.45 ! [X: poly_real] :
% 0.18/0.45 ( ( zero_zero_poly_real = X )
% 0.18/0.45 = ( X = zero_zero_poly_real ) ) ).
% 0.18/0.45
% 0.18/0.45 % zero_reorient
% 0.18/0.45 thf(fact_36_zero__reorient,axiom,
% 0.18/0.45 ! [X: int] :
% 0.18/0.45 ( ( zero_zero_int = X )
% 0.18/0.45 = ( X = zero_zero_int ) ) ).
% 0.18/0.45
% 0.18/0.45 % zero_reorient
% 0.18/0.45 thf(fact_37_zero__reorient,axiom,
% 0.18/0.45 ! [X: nat] :
% 0.18/0.45 ( ( zero_zero_nat = X )
% 0.18/0.45 = ( X = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % zero_reorient
% 0.18/0.45 thf(fact_38_zero__reorient,axiom,
% 0.18/0.45 ! [X: poly_nat] :
% 0.18/0.45 ( ( zero_zero_poly_nat = X )
% 0.18/0.45 = ( X = zero_zero_poly_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % zero_reorient
% 0.18/0.45 thf(fact_39_zero__reorient,axiom,
% 0.18/0.45 ! [X: poly_int] :
% 0.18/0.45 ( ( zero_zero_poly_int = X )
% 0.18/0.45 = ( X = zero_zero_poly_int ) ) ).
% 0.18/0.45
% 0.18/0.45 % zero_reorient
% 0.18/0.45 thf(fact_40_zero__reorient,axiom,
% 0.18/0.45 ! [X: poly_poly_real] :
% 0.18/0.45 ( ( zero_z1423781445y_real = X )
% 0.18/0.45 = ( X = zero_z1423781445y_real ) ) ).
% 0.18/0.45
% 0.18/0.45 % zero_reorient
% 0.18/0.45 thf(fact_41_poly__eq__poly__eq__iff,axiom,
% 0.18/0.45 ! [P: poly_real,Q: poly_real] :
% 0.18/0.45 ( ( ( poly_real2 @ P )
% 0.18/0.45 = ( poly_real2 @ Q ) )
% 0.18/0.45 = ( P = Q ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_eq_poly_eq_iff
% 0.18/0.45 thf(fact_42_poly__eq__poly__eq__iff,axiom,
% 0.18/0.45 ! [P: poly_int,Q: poly_int] :
% 0.18/0.45 ( ( ( poly_int2 @ P )
% 0.18/0.45 = ( poly_int2 @ Q ) )
% 0.18/0.45 = ( P = Q ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_eq_poly_eq_iff
% 0.18/0.45 thf(fact_43_poly__eq__poly__eq__iff,axiom,
% 0.18/0.45 ! [P: poly_poly_real,Q: poly_poly_real] :
% 0.18/0.45 ( ( ( poly_poly_real2 @ P )
% 0.18/0.45 = ( poly_poly_real2 @ Q ) )
% 0.18/0.45 = ( P = Q ) ) ).
% 0.18/0.45
% 0.18/0.45 % poly_eq_poly_eq_iff
% 0.18/0.45 thf(fact_44_cross__0,axiom,
% 0.18/0.45 ! [A: real,B: real] :
% 0.18/0.45 ( ( sturm_1953858694_cross @ zero_zero_poly_real @ A @ B )
% 0.18/0.45 = zero_zero_int ) ).
% 0.18/0.45
% 0.18/0.45 % cross_0
% 0.18/0.45 thf(fact_45_rsquarefree__roots,axiom,
% 0.18/0.45 ( rsquarefree_real
% 0.18/0.45 = ( ^ [P2: poly_real] :
% 0.18/0.45 ! [A2: real] :
% 0.18/0.45 ~ ( ( ( poly_real2 @ P2 @ A2 )
% 0.18/0.45 = zero_zero_real )
% 0.18/0.45 & ( ( poly_real2 @ ( pderiv_real @ P2 ) @ A2 )
% 0.18/0.45 = zero_zero_real ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rsquarefree_roots
% 0.18/0.45 thf(fact_46_cross__no__root,axiom,
% 0.18/0.45 ! [A: real,B: real,P: poly_real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( ! [X2: real] :
% 0.18/0.45 ( ( ( ord_less_real @ A @ X2 )
% 0.18/0.45 & ( ord_less_real @ X2 @ B ) )
% 0.18/0.45 => ( ( poly_real2 @ P @ X2 )
% 0.18/0.45 != zero_zero_real ) )
% 0.18/0.45 => ( ( sturm_1953858694_cross @ P @ A @ B )
% 0.18/0.45 = zero_zero_int ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % cross_no_root
% 0.18/0.45 thf(fact_47_is__zero__null,axiom,
% 0.18/0.45 ( is_zero_real
% 0.18/0.45 = ( ^ [P2: poly_real] : ( P2 = zero_zero_poly_real ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % is_zero_null
% 0.18/0.45 thf(fact_48_is__zero__null,axiom,
% 0.18/0.45 ( is_zero_nat
% 0.18/0.45 = ( ^ [P2: poly_nat] : ( P2 = zero_zero_poly_nat ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % is_zero_null
% 0.18/0.45 thf(fact_49_is__zero__null,axiom,
% 0.18/0.45 ( is_zero_int
% 0.18/0.45 = ( ^ [P2: poly_int] : ( P2 = zero_zero_poly_int ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % is_zero_null
% 0.18/0.45 thf(fact_50_is__zero__null,axiom,
% 0.18/0.45 ( is_zero_poly_real
% 0.18/0.45 = ( ^ [P2: poly_poly_real] : ( P2 = zero_z1423781445y_real ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % is_zero_null
% 0.18/0.45 thf(fact_51_poly__cutoff__0,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( poly_cutoff_real @ N @ zero_zero_poly_real )
% 0.18/0.45 = zero_zero_poly_real ) ).
% 0.18/0.45
% 0.18/0.45 % poly_cutoff_0
% 0.18/0.45 thf(fact_52_poly__cutoff__0,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
% 0.18/0.45 = zero_zero_poly_nat ) ).
% 0.18/0.45
% 0.18/0.45 % poly_cutoff_0
% 0.18/0.45 thf(fact_53_poly__cutoff__0,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( poly_cutoff_int @ N @ zero_zero_poly_int )
% 0.18/0.45 = zero_zero_poly_int ) ).
% 0.18/0.45
% 0.18/0.45 % poly_cutoff_0
% 0.18/0.45 thf(fact_54_poly__cutoff__0,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( poly_c1404107022y_real @ N @ zero_z1423781445y_real )
% 0.18/0.45 = zero_z1423781445y_real ) ).
% 0.18/0.45
% 0.18/0.45 % poly_cutoff_0
% 0.18/0.45 thf(fact_55_reflect__poly__at__0__eq__0__iff,axiom,
% 0.18/0.45 ! [P: poly_real] :
% 0.18/0.45 ( ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
% 0.18/0.45 = zero_zero_real )
% 0.18/0.45 = ( P = zero_zero_poly_real ) ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_at_0_eq_0_iff
% 0.18/0.45 thf(fact_56_reflect__poly__at__0__eq__0__iff,axiom,
% 0.18/0.45 ! [P: poly_poly_real] :
% 0.18/0.45 ( ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
% 0.18/0.45 = zero_zero_poly_real )
% 0.18/0.45 = ( P = zero_z1423781445y_real ) ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_at_0_eq_0_iff
% 0.18/0.45 thf(fact_57_reflect__poly__at__0__eq__0__iff,axiom,
% 0.18/0.45 ! [P: poly_int] :
% 0.18/0.45 ( ( ( poly_int2 @ ( reflect_poly_int @ P ) @ zero_zero_int )
% 0.18/0.45 = zero_zero_int )
% 0.18/0.45 = ( P = zero_zero_poly_int ) ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_at_0_eq_0_iff
% 0.18/0.45 thf(fact_58_reflect__poly__at__0__eq__0__iff,axiom,
% 0.18/0.45 ! [P: poly_nat] :
% 0.18/0.45 ( ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 0.18/0.45 = zero_zero_nat )
% 0.18/0.45 = ( P = zero_zero_poly_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_at_0_eq_0_iff
% 0.18/0.45 thf(fact_59_reflect__poly__at__0__eq__0__iff,axiom,
% 0.18/0.45 ! [P: poly_poly_nat] :
% 0.18/0.45 ( ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
% 0.18/0.45 = zero_zero_poly_nat )
% 0.18/0.45 = ( P = zero_z1059985641ly_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_at_0_eq_0_iff
% 0.18/0.45 thf(fact_60_reflect__poly__at__0__eq__0__iff,axiom,
% 0.18/0.45 ! [P: poly_poly_int] :
% 0.18/0.45 ( ( ( poly_poly_int2 @ ( reflec943087998ly_int @ P ) @ zero_zero_poly_int )
% 0.18/0.45 = zero_zero_poly_int )
% 0.18/0.45 = ( P = zero_z1549157189ly_int ) ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_at_0_eq_0_iff
% 0.18/0.45 thf(fact_61_reflect__poly__at__0__eq__0__iff,axiom,
% 0.18/0.45 ! [P: poly_poly_poly_real] :
% 0.18/0.45 ( ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
% 0.18/0.45 = zero_z1423781445y_real )
% 0.18/0.45 = ( P = zero_z935034829y_real ) ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_at_0_eq_0_iff
% 0.18/0.45 thf(fact_62_next__non__root__interval,axiom,
% 0.18/0.45 ! [P: poly_real,Lb: real] :
% 0.18/0.45 ( ( P != zero_zero_poly_real )
% 0.18/0.45 => ~ ! [Ub: real] :
% 0.18/0.45 ( ( ord_less_real @ Lb @ Ub )
% 0.18/0.45 => ~ ! [Z: real] :
% 0.18/0.45 ( ( ( ord_less_real @ Lb @ Z )
% 0.18/0.45 & ( ord_less_eq_real @ Z @ Ub ) )
% 0.18/0.45 => ( ( poly_real2 @ P @ Z )
% 0.18/0.45 != zero_zero_real ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % next_non_root_interval
% 0.18/0.45 thf(fact_63_last__non__root__interval,axiom,
% 0.18/0.45 ! [P: poly_real,Ub2: real] :
% 0.18/0.45 ( ( P != zero_zero_poly_real )
% 0.18/0.45 => ~ ! [Lb2: real] :
% 0.18/0.45 ( ( ord_less_real @ Lb2 @ Ub2 )
% 0.18/0.45 => ~ ! [Z: real] :
% 0.18/0.45 ( ( ( ord_less_eq_real @ Lb2 @ Z )
% 0.18/0.45 & ( ord_less_real @ Z @ Ub2 ) )
% 0.18/0.45 => ( ( poly_real2 @ P @ Z )
% 0.18/0.45 != zero_zero_real ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % last_non_root_interval
% 0.18/0.45 thf(fact_64_poly__shift__0,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( poly_shift_real @ N @ zero_zero_poly_real )
% 0.18/0.45 = zero_zero_poly_real ) ).
% 0.18/0.45
% 0.18/0.45 % poly_shift_0
% 0.18/0.45 thf(fact_65_poly__shift__0,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( poly_shift_nat @ N @ zero_zero_poly_nat )
% 0.18/0.45 = zero_zero_poly_nat ) ).
% 0.18/0.45
% 0.18/0.45 % poly_shift_0
% 0.18/0.45 thf(fact_66_poly__shift__0,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( poly_shift_int @ N @ zero_zero_poly_int )
% 0.18/0.45 = zero_zero_poly_int ) ).
% 0.18/0.45
% 0.18/0.45 % poly_shift_0
% 0.18/0.45 thf(fact_67_poly__shift__0,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( poly_shift_poly_real @ N @ zero_z1423781445y_real )
% 0.18/0.45 = zero_z1423781445y_real ) ).
% 0.18/0.45
% 0.18/0.45 % poly_shift_0
% 0.18/0.45 thf(fact_68_order__root,axiom,
% 0.18/0.45 ! [P: poly_real,A: real] :
% 0.18/0.45 ( ( ( poly_real2 @ P @ A )
% 0.18/0.45 = zero_zero_real )
% 0.18/0.45 = ( ( P = zero_zero_poly_real )
% 0.18/0.45 | ( ( order_real @ A @ P )
% 0.18/0.45 != zero_zero_nat ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_root
% 0.18/0.45 thf(fact_69_order__root,axiom,
% 0.18/0.45 ! [P: poly_poly_real,A: poly_real] :
% 0.18/0.45 ( ( ( poly_poly_real2 @ P @ A )
% 0.18/0.45 = zero_zero_poly_real )
% 0.18/0.45 = ( ( P = zero_z1423781445y_real )
% 0.18/0.45 | ( ( order_poly_real @ A @ P )
% 0.18/0.45 != zero_zero_nat ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_root
% 0.18/0.45 thf(fact_70_order__root,axiom,
% 0.18/0.45 ! [P: poly_int,A: int] :
% 0.18/0.45 ( ( ( poly_int2 @ P @ A )
% 0.18/0.45 = zero_zero_int )
% 0.18/0.45 = ( ( P = zero_zero_poly_int )
% 0.18/0.45 | ( ( order_int @ A @ P )
% 0.18/0.45 != zero_zero_nat ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_root
% 0.18/0.45 thf(fact_71_order__root,axiom,
% 0.18/0.45 ! [P: poly_poly_int,A: poly_int] :
% 0.18/0.45 ( ( ( poly_poly_int2 @ P @ A )
% 0.18/0.45 = zero_zero_poly_int )
% 0.18/0.45 = ( ( P = zero_z1549157189ly_int )
% 0.18/0.45 | ( ( order_poly_int @ A @ P )
% 0.18/0.45 != zero_zero_nat ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_root
% 0.18/0.45 thf(fact_72_order__root,axiom,
% 0.18/0.45 ! [P: poly_poly_poly_real,A: poly_poly_real] :
% 0.18/0.45 ( ( ( poly_poly_poly_real2 @ P @ A )
% 0.18/0.45 = zero_z1423781445y_real )
% 0.18/0.45 = ( ( P = zero_z935034829y_real )
% 0.18/0.45 | ( ( order_poly_poly_real @ A @ P )
% 0.18/0.45 != zero_zero_nat ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_root
% 0.18/0.45 thf(fact_73_le__zero__eq,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.18/0.45 = ( N = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_zero_eq
% 0.18/0.45 thf(fact_74_reflect__poly__0,axiom,
% 0.18/0.45 ( ( reflect_poly_real @ zero_zero_poly_real )
% 0.18/0.45 = zero_zero_poly_real ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_0
% 0.18/0.45 thf(fact_75_reflect__poly__0,axiom,
% 0.18/0.45 ( ( reflect_poly_nat @ zero_zero_poly_nat )
% 0.18/0.45 = zero_zero_poly_nat ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_0
% 0.18/0.45 thf(fact_76_reflect__poly__0,axiom,
% 0.18/0.45 ( ( reflect_poly_int @ zero_zero_poly_int )
% 0.18/0.45 = zero_zero_poly_int ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_0
% 0.18/0.45 thf(fact_77_reflect__poly__0,axiom,
% 0.18/0.45 ( ( reflec1522834046y_real @ zero_z1423781445y_real )
% 0.18/0.45 = zero_z1423781445y_real ) ).
% 0.18/0.45
% 0.18/0.45 % reflect_poly_0
% 0.18/0.45 thf(fact_78_complete__real,axiom,
% 0.18/0.45 ! [S: set_real] :
% 0.18/0.45 ( ? [X4: real] : ( member_real @ X4 @ S )
% 0.18/0.45 => ( ? [Z: real] :
% 0.18/0.45 ! [X2: real] :
% 0.18/0.45 ( ( member_real @ X2 @ S )
% 0.18/0.45 => ( ord_less_eq_real @ X2 @ Z ) )
% 0.18/0.45 => ? [Y: real] :
% 0.18/0.45 ( ! [X4: real] :
% 0.18/0.45 ( ( member_real @ X4 @ S )
% 0.18/0.45 => ( ord_less_eq_real @ X4 @ Y ) )
% 0.18/0.45 & ! [Z: real] :
% 0.18/0.45 ( ! [X2: real] :
% 0.18/0.45 ( ( member_real @ X2 @ S )
% 0.18/0.45 => ( ord_less_eq_real @ X2 @ Z ) )
% 0.18/0.45 => ( ord_less_eq_real @ Y @ Z ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % complete_real
% 0.18/0.45 thf(fact_79_le__numeral__extra_I3_J,axiom,
% 0.18/0.45 ord_le1180086932y_real @ zero_zero_poly_real @ zero_zero_poly_real ).
% 0.18/0.45
% 0.18/0.45 % le_numeral_extra(3)
% 0.18/0.45 thf(fact_80_le__numeral__extra_I3_J,axiom,
% 0.18/0.45 ord_less_eq_poly_int @ zero_zero_poly_int @ zero_zero_poly_int ).
% 0.18/0.45
% 0.18/0.45 % le_numeral_extra(3)
% 0.18/0.45 thf(fact_81_le__numeral__extra_I3_J,axiom,
% 0.18/0.45 ord_le893774876y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ).
% 0.18/0.45
% 0.18/0.45 % le_numeral_extra(3)
% 0.18/0.45 thf(fact_82_le__numeral__extra_I3_J,axiom,
% 0.18/0.45 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.18/0.45
% 0.18/0.45 % le_numeral_extra(3)
% 0.18/0.45 thf(fact_83_le__numeral__extra_I3_J,axiom,
% 0.18/0.45 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.18/0.45
% 0.18/0.45 % le_numeral_extra(3)
% 0.18/0.45 thf(fact_84_le__numeral__extra_I3_J,axiom,
% 0.18/0.45 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.18/0.45
% 0.18/0.45 % le_numeral_extra(3)
% 0.18/0.45 thf(fact_85_zero__le,axiom,
% 0.18/0.45 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.18/0.45
% 0.18/0.45 % zero_le
% 0.18/0.45 thf(fact_86_less__eq__real__def,axiom,
% 0.18/0.45 ( ord_less_eq_real
% 0.18/0.45 = ( ^ [X3: real,Y2: real] :
% 0.18/0.45 ( ( ord_less_real @ X3 @ Y2 )
% 0.18/0.45 | ( X3 = Y2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_eq_real_def
% 0.18/0.45 thf(fact_87_order__0I,axiom,
% 0.18/0.45 ! [P: poly_real,A: real] :
% 0.18/0.45 ( ( ( poly_real2 @ P @ A )
% 0.18/0.45 != zero_zero_real )
% 0.18/0.45 => ( ( order_real @ A @ P )
% 0.18/0.45 = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_0I
% 0.18/0.45 thf(fact_88_order__0I,axiom,
% 0.18/0.45 ! [P: poly_poly_real,A: poly_real] :
% 0.18/0.45 ( ( ( poly_poly_real2 @ P @ A )
% 0.18/0.45 != zero_zero_poly_real )
% 0.18/0.45 => ( ( order_poly_real @ A @ P )
% 0.18/0.45 = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_0I
% 0.18/0.45 thf(fact_89_order__0I,axiom,
% 0.18/0.45 ! [P: poly_int,A: int] :
% 0.18/0.45 ( ( ( poly_int2 @ P @ A )
% 0.18/0.45 != zero_zero_int )
% 0.18/0.45 => ( ( order_int @ A @ P )
% 0.18/0.45 = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_0I
% 0.18/0.45 thf(fact_90_order__0I,axiom,
% 0.18/0.45 ! [P: poly_poly_int,A: poly_int] :
% 0.18/0.45 ( ( ( poly_poly_int2 @ P @ A )
% 0.18/0.45 != zero_zero_poly_int )
% 0.18/0.45 => ( ( order_poly_int @ A @ P )
% 0.18/0.45 = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_0I
% 0.18/0.45 thf(fact_91_order__0I,axiom,
% 0.18/0.45 ! [P: poly_poly_poly_real,A: poly_poly_real] :
% 0.18/0.45 ( ( ( poly_poly_poly_real2 @ P @ A )
% 0.18/0.45 != zero_z1423781445y_real )
% 0.18/0.45 => ( ( order_poly_poly_real @ A @ P )
% 0.18/0.45 = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_0I
% 0.18/0.45 thf(fact_92_not__eq__pos__or__neg__iff__1,axiom,
% 0.18/0.45 ! [Lb: real,Ub2: real,P: poly_real] :
% 0.18/0.45 ( ( ! [Z2: real] :
% 0.18/0.45 ( ( ( ord_less_real @ Lb @ Z2 )
% 0.18/0.45 & ( ord_less_eq_real @ Z2 @ Ub2 ) )
% 0.18/0.45 => ( ( poly_real2 @ P @ Z2 )
% 0.18/0.45 != zero_zero_real ) ) )
% 0.18/0.45 = ( ! [Z2: real] :
% 0.18/0.45 ( ( ( ord_less_real @ Lb @ Z2 )
% 0.18/0.45 & ( ord_less_eq_real @ Z2 @ Ub2 ) )
% 0.18/0.45 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z2 ) ) )
% 0.18/0.45 | ! [Z2: real] :
% 0.18/0.45 ( ( ( ord_less_real @ Lb @ Z2 )
% 0.18/0.45 & ( ord_less_eq_real @ Z2 @ Ub2 ) )
% 0.18/0.45 => ( ord_less_real @ ( poly_real2 @ P @ Z2 ) @ zero_zero_real ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % not_eq_pos_or_neg_iff_1
% 0.18/0.45 thf(fact_93_not__eq__pos__or__neg__iff__2,axiom,
% 0.18/0.45 ! [Lb: real,Ub2: real,P: poly_real] :
% 0.18/0.45 ( ( ! [Z2: real] :
% 0.18/0.45 ( ( ( ord_less_eq_real @ Lb @ Z2 )
% 0.18/0.45 & ( ord_less_real @ Z2 @ Ub2 ) )
% 0.18/0.45 => ( ( poly_real2 @ P @ Z2 )
% 0.18/0.45 != zero_zero_real ) ) )
% 0.18/0.45 = ( ! [Z2: real] :
% 0.18/0.45 ( ( ( ord_less_eq_real @ Lb @ Z2 )
% 0.18/0.45 & ( ord_less_real @ Z2 @ Ub2 ) )
% 0.18/0.45 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z2 ) ) )
% 0.18/0.45 | ! [Z2: real] :
% 0.18/0.45 ( ( ( ord_less_eq_real @ Lb @ Z2 )
% 0.18/0.45 & ( ord_less_real @ Z2 @ Ub2 ) )
% 0.18/0.45 => ( ord_less_real @ ( poly_real2 @ P @ Z2 ) @ zero_zero_real ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % not_eq_pos_or_neg_iff_2
% 0.18/0.45 thf(fact_94_neq0__conv,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( N != zero_zero_nat )
% 0.18/0.45 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % neq0_conv
% 0.18/0.45 thf(fact_95_less__nat__zero__code,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.18/0.45
% 0.18/0.45 % less_nat_zero_code
% 0.18/0.45 thf(fact_96_bot__nat__0_Onot__eq__extremum,axiom,
% 0.18/0.45 ! [A: nat] :
% 0.18/0.45 ( ( A != zero_zero_nat )
% 0.18/0.45 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % bot_nat_0.not_eq_extremum
% 0.18/0.45 thf(fact_97_order__refl,axiom,
% 0.18/0.45 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % order_refl
% 0.18/0.45 thf(fact_98_order__refl,axiom,
% 0.18/0.45 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % order_refl
% 0.18/0.45 thf(fact_99_order__refl,axiom,
% 0.18/0.45 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % order_refl
% 0.18/0.45 thf(fact_100_minf_I8_J,axiom,
% 0.18/0.45 ! [T: real] :
% 0.18/0.45 ? [Z3: real] :
% 0.18/0.45 ! [X4: real] :
% 0.18/0.45 ( ( ord_less_real @ X4 @ Z3 )
% 0.18/0.45 => ~ ( ord_less_eq_real @ T @ X4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % minf(8)
% 0.18/0.45 thf(fact_101_minf_I8_J,axiom,
% 0.18/0.45 ! [T: nat] :
% 0.18/0.45 ? [Z3: nat] :
% 0.18/0.45 ! [X4: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X4 @ Z3 )
% 0.18/0.45 => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % minf(8)
% 0.18/0.45 thf(fact_102_minf_I8_J,axiom,
% 0.18/0.45 ! [T: int] :
% 0.18/0.45 ? [Z3: int] :
% 0.18/0.45 ! [X4: int] :
% 0.18/0.45 ( ( ord_less_int @ X4 @ Z3 )
% 0.18/0.45 => ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % minf(8)
% 0.18/0.45 thf(fact_103_minf_I6_J,axiom,
% 0.18/0.45 ! [T: real] :
% 0.18/0.45 ? [Z3: real] :
% 0.18/0.45 ! [X4: real] :
% 0.18/0.45 ( ( ord_less_real @ X4 @ Z3 )
% 0.18/0.45 => ( ord_less_eq_real @ X4 @ T ) ) ).
% 0.18/0.45
% 0.18/0.45 % minf(6)
% 0.18/0.45 thf(fact_104_minf_I6_J,axiom,
% 0.18/0.45 ! [T: nat] :
% 0.18/0.45 ? [Z3: nat] :
% 0.18/0.45 ! [X4: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X4 @ Z3 )
% 0.18/0.45 => ( ord_less_eq_nat @ X4 @ T ) ) ).
% 0.18/0.45
% 0.18/0.45 % minf(6)
% 0.18/0.45 thf(fact_105_minf_I6_J,axiom,
% 0.18/0.45 ! [T: int] :
% 0.18/0.45 ? [Z3: int] :
% 0.18/0.45 ! [X4: int] :
% 0.18/0.45 ( ( ord_less_int @ X4 @ Z3 )
% 0.18/0.45 => ( ord_less_eq_int @ X4 @ T ) ) ).
% 0.18/0.45
% 0.18/0.45 % minf(6)
% 0.18/0.45 thf(fact_106_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: real,P3: real > $o] :
% 0.18/0.45 ( ( member_real @ A @ ( collect_real @ P3 ) )
% 0.18/0.45 = ( P3 @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_107_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A3: set_real] :
% 0.18/0.45 ( ( collect_real
% 0.18/0.45 @ ^ [X3: real] : ( member_real @ X3 @ A3 ) )
% 0.18/0.45 = A3 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_108_pinf_I8_J,axiom,
% 0.18/0.45 ! [T: real] :
% 0.18/0.45 ? [Z3: real] :
% 0.18/0.45 ! [X4: real] :
% 0.18/0.45 ( ( ord_less_real @ Z3 @ X4 )
% 0.18/0.45 => ( ord_less_eq_real @ T @ X4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % pinf(8)
% 0.18/0.45 thf(fact_109_pinf_I8_J,axiom,
% 0.18/0.45 ! [T: nat] :
% 0.18/0.45 ? [Z3: nat] :
% 0.18/0.45 ! [X4: nat] :
% 0.18/0.45 ( ( ord_less_nat @ Z3 @ X4 )
% 0.18/0.45 => ( ord_less_eq_nat @ T @ X4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % pinf(8)
% 0.18/0.45 thf(fact_110_pinf_I8_J,axiom,
% 0.18/0.45 ! [T: int] :
% 0.18/0.45 ? [Z3: int] :
% 0.18/0.45 ! [X4: int] :
% 0.18/0.45 ( ( ord_less_int @ Z3 @ X4 )
% 0.18/0.45 => ( ord_less_eq_int @ T @ X4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % pinf(8)
% 0.18/0.45 thf(fact_111_pinf_I6_J,axiom,
% 0.18/0.45 ! [T: real] :
% 0.18/0.45 ? [Z3: real] :
% 0.18/0.45 ! [X4: real] :
% 0.18/0.45 ( ( ord_less_real @ Z3 @ X4 )
% 0.18/0.45 => ~ ( ord_less_eq_real @ X4 @ T ) ) ).
% 0.18/0.45
% 0.18/0.45 % pinf(6)
% 0.18/0.45 thf(fact_112_pinf_I6_J,axiom,
% 0.18/0.45 ! [T: nat] :
% 0.18/0.45 ? [Z3: nat] :
% 0.18/0.45 ! [X4: nat] :
% 0.18/0.45 ( ( ord_less_nat @ Z3 @ X4 )
% 0.18/0.45 => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% 0.18/0.45
% 0.18/0.45 % pinf(6)
% 0.18/0.45 thf(fact_113_pinf_I6_J,axiom,
% 0.18/0.45 ! [T: int] :
% 0.18/0.45 ? [Z3: int] :
% 0.18/0.45 ! [X4: int] :
% 0.18/0.45 ( ( ord_less_int @ Z3 @ X4 )
% 0.18/0.45 => ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% 0.18/0.45
% 0.18/0.45 % pinf(6)
% 0.18/0.45 thf(fact_114_verit__comp__simplify1_I3_J,axiom,
% 0.18/0.45 ! [B2: real,A4: real] :
% 0.18/0.45 ( ( ~ ( ord_less_eq_real @ B2 @ A4 ) )
% 0.18/0.45 = ( ord_less_real @ A4 @ B2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_comp_simplify1(3)
% 0.18/0.45 thf(fact_115_verit__comp__simplify1_I3_J,axiom,
% 0.18/0.45 ! [B2: nat,A4: nat] :
% 0.18/0.45 ( ( ~ ( ord_less_eq_nat @ B2 @ A4 ) )
% 0.18/0.45 = ( ord_less_nat @ A4 @ B2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_comp_simplify1(3)
% 0.18/0.45 thf(fact_116_verit__comp__simplify1_I3_J,axiom,
% 0.18/0.45 ! [B2: int,A4: int] :
% 0.18/0.45 ( ( ~ ( ord_less_eq_int @ B2 @ A4 ) )
% 0.18/0.45 = ( ord_less_int @ A4 @ B2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_comp_simplify1(3)
% 0.18/0.45 thf(fact_117_bot__nat__0_Oextremum,axiom,
% 0.18/0.45 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.18/0.45
% 0.18/0.45 % bot_nat_0.extremum
% 0.18/0.45 thf(fact_118_le0,axiom,
% 0.18/0.45 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.18/0.45
% 0.18/0.45 % le0
% 0.18/0.45 thf(fact_119_nat__less__le,axiom,
% 0.18/0.45 ( ord_less_nat
% 0.18/0.45 = ( ^ [M2: nat,N2: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.18/0.45 & ( M2 != N2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % nat_less_le
% 0.18/0.45 thf(fact_120_less__imp__le__nat,axiom,
% 0.18/0.45 ! [M: nat,N: nat] :
% 0.18/0.45 ( ( ord_less_nat @ M @ N )
% 0.18/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_le_nat
% 0.18/0.45 thf(fact_121_le__eq__less__or__eq,axiom,
% 0.18/0.45 ( ord_less_eq_nat
% 0.18/0.45 = ( ^ [M2: nat,N2: nat] :
% 0.18/0.45 ( ( ord_less_nat @ M2 @ N2 )
% 0.18/0.45 | ( M2 = N2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_eq_less_or_eq
% 0.18/0.45 thf(fact_122_less__or__eq__imp__le,axiom,
% 0.18/0.45 ! [M: nat,N: nat] :
% 0.18/0.45 ( ( ( ord_less_nat @ M @ N )
% 0.18/0.45 | ( M = N ) )
% 0.18/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_or_eq_imp_le
% 0.18/0.45 thf(fact_123_le__neq__implies__less,axiom,
% 0.18/0.45 ! [M: nat,N: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.45 => ( ( M != N )
% 0.18/0.45 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_neq_implies_less
% 0.18/0.45 thf(fact_124_less__mono__imp__le__mono,axiom,
% 0.18/0.45 ! [F: nat > nat,I: nat,J: nat] :
% 0.18/0.45 ( ! [I2: nat,J2: nat] :
% 0.18/0.45 ( ( ord_less_nat @ I2 @ J2 )
% 0.18/0.45 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.18/0.45 => ( ( ord_less_eq_nat @ I @ J )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_mono_imp_le_mono
% 0.18/0.45 thf(fact_125_linorder__neqE__nat,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( X != Y3 )
% 0.18/0.45 => ( ~ ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % linorder_neqE_nat
% 0.18/0.45 thf(fact_126_infinite__descent,axiom,
% 0.18/0.45 ! [P3: nat > $o,N: nat] :
% 0.18/0.45 ( ! [N3: nat] :
% 0.18/0.45 ( ~ ( P3 @ N3 )
% 0.18/0.45 => ? [M3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ M3 @ N3 )
% 0.18/0.45 & ~ ( P3 @ M3 ) ) )
% 0.18/0.45 => ( P3 @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % infinite_descent
% 0.18/0.45 thf(fact_127_nat__less__induct,axiom,
% 0.18/0.45 ! [P3: nat > $o,N: nat] :
% 0.18/0.45 ( ! [N3: nat] :
% 0.18/0.45 ( ! [M3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ M3 @ N3 )
% 0.18/0.45 => ( P3 @ M3 ) )
% 0.18/0.45 => ( P3 @ N3 ) )
% 0.18/0.45 => ( P3 @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % nat_less_induct
% 0.18/0.45 thf(fact_128_less__irrefl__nat,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ~ ( ord_less_nat @ N @ N ) ).
% 0.18/0.45
% 0.18/0.45 % less_irrefl_nat
% 0.18/0.45 thf(fact_129_less__not__refl3,axiom,
% 0.18/0.45 ! [S2: nat,T: nat] :
% 0.18/0.45 ( ( ord_less_nat @ S2 @ T )
% 0.18/0.45 => ( S2 != T ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_not_refl3
% 0.18/0.45 thf(fact_130_less__not__refl2,axiom,
% 0.18/0.45 ! [N: nat,M: nat] :
% 0.18/0.45 ( ( ord_less_nat @ N @ M )
% 0.18/0.45 => ( M != N ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_not_refl2
% 0.18/0.45 thf(fact_131_less__not__refl,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ~ ( ord_less_nat @ N @ N ) ).
% 0.18/0.45
% 0.18/0.45 % less_not_refl
% 0.18/0.45 thf(fact_132_nat__neq__iff,axiom,
% 0.18/0.45 ! [M: nat,N: nat] :
% 0.18/0.45 ( ( M != N )
% 0.18/0.45 = ( ( ord_less_nat @ M @ N )
% 0.18/0.45 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % nat_neq_iff
% 0.18/0.45 thf(fact_133_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.18/0.45 ! [A: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.18/0.45 => ( A = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % bot_nat_0.extremum_uniqueI
% 0.18/0.45 thf(fact_134_bot__nat__0_Oextremum__unique,axiom,
% 0.18/0.45 ! [A: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.18/0.45 = ( A = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % bot_nat_0.extremum_unique
% 0.18/0.45 thf(fact_135_ex__least__nat__le,axiom,
% 0.18/0.45 ! [P3: nat > $o,N: nat] :
% 0.18/0.45 ( ( P3 @ N )
% 0.18/0.45 => ( ~ ( P3 @ zero_zero_nat )
% 0.18/0.45 => ? [K: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ K @ N )
% 0.18/0.45 & ! [I3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ I3 @ K )
% 0.18/0.45 => ~ ( P3 @ I3 ) )
% 0.18/0.45 & ( P3 @ K ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ex_least_nat_le
% 0.18/0.45 thf(fact_136_le__0__eq,axiom,
% 0.18/0.45 ! [N: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.18/0.45 = ( N = zero_zero_nat ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_0_eq
% 0.18/0.45 thf(fact_137_less__eq__nat_Osimps_I1_J,axiom,
% 0.18/0.45 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.18/0.45
% 0.18/0.45 % less_eq_nat.simps(1)
% 0.18/0.45 thf(fact_138_conj__le__cong,axiom,
% 0.18/0.45 ! [X: int,X5: int,P3: $o,P4: $o] :
% 0.18/0.45 ( ( X = X5 )
% 0.18/0.45 => ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 0.18/0.45 => ( P3 = P4 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.18/0.45 & P3 )
% 0.18/0.45 = ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 0.18/0.45 & P4 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % conj_le_cong
% 0.18/0.45 thf(fact_139_imp__le__cong,axiom,
% 0.18/0.45 ! [X: int,X5: int,P3: $o,P4: $o] :
% 0.18/0.45 ( ( X = X5 )
% 0.18/0.45 => ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 0.18/0.45 => ( P3 = P4 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.18/0.45 => P3 )
% 0.18/0.45 = ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 0.18/0.45 => P4 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % imp_le_cong
% 0.18/0.45 thf(fact_140_order__subst1,axiom,
% 0.18/0.45 ! [A: real,F: real > real,B: real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst1
% 0.18/0.45 thf(fact_141_order__subst1,axiom,
% 0.18/0.45 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst1
% 0.18/0.45 thf(fact_142_order__subst1,axiom,
% 0.18/0.45 ! [A: real,F: int > real,B: int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst1
% 0.18/0.45 thf(fact_143_order__subst1,axiom,
% 0.18/0.45 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst1
% 0.18/0.45 thf(fact_144_order__subst1,axiom,
% 0.18/0.45 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst1
% 0.18/0.45 thf(fact_145_order__subst1,axiom,
% 0.18/0.45 ! [A: nat,F: int > nat,B: int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst1
% 0.18/0.45 thf(fact_146_order__subst1,axiom,
% 0.18/0.45 ! [A: int,F: real > int,B: real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst1
% 0.18/0.45 thf(fact_147_order__subst1,axiom,
% 0.18/0.45 ! [A: int,F: nat > int,B: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst1
% 0.18/0.45 thf(fact_148_order__subst1,axiom,
% 0.18/0.45 ! [A: int,F: int > int,B: int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst1
% 0.18/0.45 thf(fact_149_order__subst2,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst2
% 0.18/0.45 thf(fact_150_order__subst2,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst2
% 0.18/0.45 thf(fact_151_order__subst2,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst2
% 0.18/0.45 thf(fact_152_order__subst2,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst2
% 0.18/0.45 thf(fact_153_order__subst2,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst2
% 0.18/0.45 thf(fact_154_order__subst2,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst2
% 0.18/0.45 thf(fact_155_order__subst2,axiom,
% 0.18/0.45 ! [A: int,B: int,F: int > real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst2
% 0.18/0.45 thf(fact_156_order__subst2,axiom,
% 0.18/0.45 ! [A: int,B: int,F: int > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst2
% 0.18/0.45 thf(fact_157_order__subst2,axiom,
% 0.18/0.45 ! [A: int,B: int,F: int > int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_subst2
% 0.18/0.45 thf(fact_158_verit__la__disequality,axiom,
% 0.18/0.45 ! [A: real,B: real] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 | ~ ( ord_less_eq_real @ A @ B )
% 0.18/0.45 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_la_disequality
% 0.18/0.45 thf(fact_159_verit__la__disequality,axiom,
% 0.18/0.45 ! [A: nat,B: nat] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 | ~ ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_la_disequality
% 0.18/0.45 thf(fact_160_verit__la__disequality,axiom,
% 0.18/0.45 ! [A: int,B: int] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 | ~ ( ord_less_eq_int @ A @ B )
% 0.18/0.45 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_la_disequality
% 0.18/0.45 thf(fact_161_ord__eq__le__subst,axiom,
% 0.18/0.45 ! [A: real,F: real > real,B: real,C: real] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_subst
% 0.18/0.45 thf(fact_162_ord__eq__le__subst,axiom,
% 0.18/0.45 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_subst
% 0.18/0.45 thf(fact_163_ord__eq__le__subst,axiom,
% 0.18/0.45 ! [A: int,F: real > int,B: real,C: real] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_subst
% 0.18/0.45 thf(fact_164_ord__eq__le__subst,axiom,
% 0.18/0.45 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_subst
% 0.18/0.45 thf(fact_165_ord__eq__le__subst,axiom,
% 0.18/0.45 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_subst
% 0.18/0.45 thf(fact_166_ord__eq__le__subst,axiom,
% 0.18/0.45 ! [A: int,F: nat > int,B: nat,C: nat] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_subst
% 0.18/0.45 thf(fact_167_ord__eq__le__subst,axiom,
% 0.18/0.45 ! [A: real,F: int > real,B: int,C: int] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_subst
% 0.18/0.45 thf(fact_168_ord__eq__le__subst,axiom,
% 0.18/0.45 ! [A: nat,F: int > nat,B: int,C: int] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_subst
% 0.18/0.45 thf(fact_169_ord__eq__le__subst,axiom,
% 0.18/0.45 ! [A: int,F: int > int,B: int,C: int] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_subst
% 0.18/0.45 thf(fact_170_ord__le__eq__subst,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_subst
% 0.18/0.45 thf(fact_171_ord__le__eq__subst,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_subst
% 0.18/0.45 thf(fact_172_ord__le__eq__subst,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_subst
% 0.18/0.45 thf(fact_173_ord__le__eq__subst,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_subst
% 0.18/0.45 thf(fact_174_ord__le__eq__subst,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_subst
% 0.18/0.45 thf(fact_175_ord__le__eq__subst,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_subst
% 0.18/0.45 thf(fact_176_ord__le__eq__subst,axiom,
% 0.18/0.45 ! [A: int,B: int,F: int > real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_subst
% 0.18/0.45 thf(fact_177_ord__le__eq__subst,axiom,
% 0.18/0.45 ! [A: int,B: int,F: int > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_subst
% 0.18/0.45 thf(fact_178_ord__le__eq__subst,axiom,
% 0.18/0.45 ! [A: int,B: int,F: int > int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X2 @ Y )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_subst
% 0.18/0.45 thf(fact_179_eq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y4: real,Z4: real] : ( Y4 = Z4 ) )
% 0.18/0.45 = ( ^ [X3: real,Y2: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.18/0.45 & ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % eq_iff
% 0.18/0.45 thf(fact_180_eq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 0.18/0.45 = ( ^ [X3: nat,Y2: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.18/0.45 & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % eq_iff
% 0.18/0.45 thf(fact_181_eq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 0.18/0.45 = ( ^ [X3: int,Y2: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.18/0.45 & ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % eq_iff
% 0.18/0.45 thf(fact_182_antisym,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_eq_real @ Y3 @ X )
% 0.18/0.45 => ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % antisym
% 0.18/0.45 thf(fact_183_antisym,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_eq_nat @ Y3 @ X )
% 0.18/0.45 => ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % antisym
% 0.18/0.45 thf(fact_184_antisym,axiom,
% 0.18/0.45 ! [X: int,Y3: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_eq_int @ Y3 @ X )
% 0.18/0.45 => ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % antisym
% 0.18/0.45 thf(fact_185_linear,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.18/0.45 | ( ord_less_eq_real @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % linear
% 0.18/0.45 thf(fact_186_linear,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.18/0.45 | ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % linear
% 0.18/0.45 thf(fact_187_linear,axiom,
% 0.18/0.45 ! [X: int,Y3: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X @ Y3 )
% 0.18/0.45 | ( ord_less_eq_int @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % linear
% 0.18/0.45 thf(fact_188_eq__refl,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( X = Y3 )
% 0.18/0.45 => ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % eq_refl
% 0.18/0.45 thf(fact_189_eq__refl,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( X = Y3 )
% 0.18/0.45 => ( ord_less_eq_nat @ X @ Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % eq_refl
% 0.18/0.45 thf(fact_190_eq__refl,axiom,
% 0.18/0.45 ! [X: int,Y3: int] :
% 0.18/0.45 ( ( X = Y3 )
% 0.18/0.45 => ( ord_less_eq_int @ X @ Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % eq_refl
% 0.18/0.45 thf(fact_191_le__cases,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ~ ( ord_less_eq_real @ X @ Y3 )
% 0.18/0.45 => ( ord_less_eq_real @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_cases
% 0.18/0.45 thf(fact_192_le__cases,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ~ ( ord_less_eq_nat @ X @ Y3 )
% 0.18/0.45 => ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_cases
% 0.18/0.45 thf(fact_193_le__cases,axiom,
% 0.18/0.45 ! [X: int,Y3: int] :
% 0.18/0.45 ( ~ ( ord_less_eq_int @ X @ Y3 )
% 0.18/0.45 => ( ord_less_eq_int @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_cases
% 0.18/0.45 thf(fact_194_order_Otrans,axiom,
% 0.18/0.45 ! [A: real,B: real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.45 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order.trans
% 0.18/0.45 thf(fact_195_order_Otrans,axiom,
% 0.18/0.45 ! [A: nat,B: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.18/0.45 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order.trans
% 0.18/0.45 thf(fact_196_order_Otrans,axiom,
% 0.18/0.45 ! [A: int,B: int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.45 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order.trans
% 0.18/0.45 thf(fact_197_le__cases3,axiom,
% 0.18/0.45 ! [X: real,Y3: real,Z5: real] :
% 0.18/0.45 ( ( ( ord_less_eq_real @ X @ Y3 )
% 0.18/0.45 => ~ ( ord_less_eq_real @ Y3 @ Z5 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_real @ Y3 @ X )
% 0.18/0.45 => ~ ( ord_less_eq_real @ X @ Z5 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_real @ X @ Z5 )
% 0.18/0.45 => ~ ( ord_less_eq_real @ Z5 @ Y3 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_real @ Z5 @ Y3 )
% 0.18/0.45 => ~ ( ord_less_eq_real @ Y3 @ X ) )
% 0.18/0.45 => ( ( ( ord_less_eq_real @ Y3 @ Z5 )
% 0.18/0.45 => ~ ( ord_less_eq_real @ Z5 @ X ) )
% 0.18/0.45 => ~ ( ( ord_less_eq_real @ Z5 @ X )
% 0.18/0.45 => ~ ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_cases3
% 0.18/0.45 thf(fact_198_le__cases3,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat,Z5: nat] :
% 0.18/0.45 ( ( ( ord_less_eq_nat @ X @ Y3 )
% 0.18/0.45 => ~ ( ord_less_eq_nat @ Y3 @ Z5 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_nat @ Y3 @ X )
% 0.18/0.45 => ~ ( ord_less_eq_nat @ X @ Z5 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_nat @ X @ Z5 )
% 0.18/0.45 => ~ ( ord_less_eq_nat @ Z5 @ Y3 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_nat @ Z5 @ Y3 )
% 0.18/0.45 => ~ ( ord_less_eq_nat @ Y3 @ X ) )
% 0.18/0.45 => ( ( ( ord_less_eq_nat @ Y3 @ Z5 )
% 0.18/0.45 => ~ ( ord_less_eq_nat @ Z5 @ X ) )
% 0.18/0.45 => ~ ( ( ord_less_eq_nat @ Z5 @ X )
% 0.18/0.45 => ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_cases3
% 0.18/0.45 thf(fact_199_le__cases3,axiom,
% 0.18/0.45 ! [X: int,Y3: int,Z5: int] :
% 0.18/0.45 ( ( ( ord_less_eq_int @ X @ Y3 )
% 0.18/0.45 => ~ ( ord_less_eq_int @ Y3 @ Z5 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_int @ Y3 @ X )
% 0.18/0.45 => ~ ( ord_less_eq_int @ X @ Z5 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_int @ X @ Z5 )
% 0.18/0.45 => ~ ( ord_less_eq_int @ Z5 @ Y3 ) )
% 0.18/0.45 => ( ( ( ord_less_eq_int @ Z5 @ Y3 )
% 0.18/0.45 => ~ ( ord_less_eq_int @ Y3 @ X ) )
% 0.18/0.45 => ( ( ( ord_less_eq_int @ Y3 @ Z5 )
% 0.18/0.45 => ~ ( ord_less_eq_int @ Z5 @ X ) )
% 0.18/0.45 => ~ ( ( ord_less_eq_int @ Z5 @ X )
% 0.18/0.45 => ~ ( ord_less_eq_int @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_cases3
% 0.18/0.45 thf(fact_200_antisym__conv,axiom,
% 0.18/0.45 ! [Y3: real,X: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ Y3 @ X )
% 0.18/0.45 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.18/0.45 = ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % antisym_conv
% 0.18/0.45 thf(fact_201_antisym__conv,axiom,
% 0.18/0.45 ! [Y3: nat,X: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ Y3 @ X )
% 0.18/0.45 => ( ( ord_less_eq_nat @ X @ Y3 )
% 0.18/0.45 = ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % antisym_conv
% 0.18/0.45 thf(fact_202_antisym__conv,axiom,
% 0.18/0.45 ! [Y3: int,X: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ Y3 @ X )
% 0.18/0.45 => ( ( ord_less_eq_int @ X @ Y3 )
% 0.18/0.45 = ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % antisym_conv
% 0.18/0.45 thf(fact_203_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y4: real,Z4: real] : ( Y4 = Z4 ) )
% 0.18/0.45 = ( ^ [A2: real,B3: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.18/0.45 & ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_class.order.eq_iff
% 0.18/0.45 thf(fact_204_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 0.18/0.45 = ( ^ [A2: nat,B3: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.18/0.45 & ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_class.order.eq_iff
% 0.18/0.45 thf(fact_205_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 0.18/0.45 = ( ^ [A2: int,B3: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.18/0.45 & ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_class.order.eq_iff
% 0.18/0.45 thf(fact_206_ord__eq__le__trans,axiom,
% 0.18/0.45 ! [A: real,B: real,C: real] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.45 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_trans
% 0.18/0.45 thf(fact_207_ord__eq__le__trans,axiom,
% 0.18/0.45 ! [A: nat,B: nat,C: nat] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.18/0.45 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_trans
% 0.18/0.45 thf(fact_208_ord__eq__le__trans,axiom,
% 0.18/0.45 ! [A: int,B: int,C: int] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.45 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_le_trans
% 0.18/0.45 thf(fact_209_ord__le__eq__trans,axiom,
% 0.18/0.45 ! [A: real,B: real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( ( B = C )
% 0.18/0.45 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_trans
% 0.18/0.45 thf(fact_210_ord__le__eq__trans,axiom,
% 0.18/0.45 ! [A: nat,B: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( ( B = C )
% 0.18/0.45 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_trans
% 0.18/0.45 thf(fact_211_ord__le__eq__trans,axiom,
% 0.18/0.45 ! [A: int,B: int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( ( B = C )
% 0.18/0.45 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_le_eq_trans
% 0.18/0.45 thf(fact_212_order__class_Oorder_Oantisym,axiom,
% 0.18/0.45 ! [A: real,B: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_real @ B @ A )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_class.order.antisym
% 0.18/0.45 thf(fact_213_order__class_Oorder_Oantisym,axiom,
% 0.18/0.45 ! [A: nat,B: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_nat @ B @ A )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_class.order.antisym
% 0.18/0.45 thf(fact_214_order__class_Oorder_Oantisym,axiom,
% 0.18/0.45 ! [A: int,B: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_int @ B @ A )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_class.order.antisym
% 0.18/0.45 thf(fact_215_order__trans,axiom,
% 0.18/0.45 ! [X: real,Y3: real,Z5: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_eq_real @ Y3 @ Z5 )
% 0.18/0.45 => ( ord_less_eq_real @ X @ Z5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_trans
% 0.18/0.45 thf(fact_216_order__trans,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat,Z5: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_eq_nat @ Y3 @ Z5 )
% 0.18/0.45 => ( ord_less_eq_nat @ X @ Z5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_trans
% 0.18/0.45 thf(fact_217_order__trans,axiom,
% 0.18/0.45 ! [X: int,Y3: int,Z5: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_eq_int @ Y3 @ Z5 )
% 0.18/0.45 => ( ord_less_eq_int @ X @ Z5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_trans
% 0.18/0.45 thf(fact_218_dual__order_Orefl,axiom,
% 0.18/0.45 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.refl
% 0.18/0.45 thf(fact_219_dual__order_Orefl,axiom,
% 0.18/0.45 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.refl
% 0.18/0.45 thf(fact_220_dual__order_Orefl,axiom,
% 0.18/0.45 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.refl
% 0.18/0.45 thf(fact_221_linorder__wlog,axiom,
% 0.18/0.45 ! [P3: real > real > $o,A: real,B: real] :
% 0.18/0.45 ( ! [A5: real,B4: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ A5 @ B4 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( ! [A5: real,B4: real] :
% 0.18/0.45 ( ( P3 @ B4 @ A5 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( P3 @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % linorder_wlog
% 0.18/0.45 thf(fact_222_linorder__wlog,axiom,
% 0.18/0.45 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 0.18/0.45 ( ! [A5: nat,B4: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ A5 @ B4 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( ! [A5: nat,B4: nat] :
% 0.18/0.45 ( ( P3 @ B4 @ A5 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( P3 @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % linorder_wlog
% 0.18/0.45 thf(fact_223_linorder__wlog,axiom,
% 0.18/0.45 ! [P3: int > int > $o,A: int,B: int] :
% 0.18/0.45 ( ! [A5: int,B4: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ A5 @ B4 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( ! [A5: int,B4: int] :
% 0.18/0.45 ( ( P3 @ B4 @ A5 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( P3 @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % linorder_wlog
% 0.18/0.45 thf(fact_224_dual__order_Otrans,axiom,
% 0.18/0.45 ! [B: real,A: real,C: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ B @ A )
% 0.18/0.45 => ( ( ord_less_eq_real @ C @ B )
% 0.18/0.45 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.trans
% 0.18/0.45 thf(fact_225_dual__order_Otrans,axiom,
% 0.18/0.45 ! [B: nat,A: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ B @ A )
% 0.18/0.45 => ( ( ord_less_eq_nat @ C @ B )
% 0.18/0.45 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.trans
% 0.18/0.45 thf(fact_226_dual__order_Otrans,axiom,
% 0.18/0.45 ! [B: int,A: int,C: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ B @ A )
% 0.18/0.45 => ( ( ord_less_eq_int @ C @ B )
% 0.18/0.45 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.trans
% 0.18/0.45 thf(fact_227_dual__order_Oeq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y4: real,Z4: real] : ( Y4 = Z4 ) )
% 0.18/0.45 = ( ^ [A2: real,B3: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ B3 @ A2 )
% 0.18/0.45 & ( ord_less_eq_real @ A2 @ B3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.eq_iff
% 0.18/0.45 thf(fact_228_dual__order_Oeq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 0.18/0.45 = ( ^ [A2: nat,B3: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.18/0.45 & ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.eq_iff
% 0.18/0.45 thf(fact_229_dual__order_Oeq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 0.18/0.45 = ( ^ [A2: int,B3: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.18/0.45 & ( ord_less_eq_int @ A2 @ B3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.eq_iff
% 0.18/0.45 thf(fact_230_dual__order_Oantisym,axiom,
% 0.18/0.45 ! [B: real,A: real] :
% 0.18/0.45 ( ( ord_less_eq_real @ B @ A )
% 0.18/0.45 => ( ( ord_less_eq_real @ A @ B )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.antisym
% 0.18/0.45 thf(fact_231_dual__order_Oantisym,axiom,
% 0.18/0.45 ! [B: nat,A: nat] :
% 0.18/0.45 ( ( ord_less_eq_nat @ B @ A )
% 0.18/0.45 => ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.antisym
% 0.18/0.45 thf(fact_232_dual__order_Oantisym,axiom,
% 0.18/0.45 ! [B: int,A: int] :
% 0.18/0.45 ( ( ord_less_eq_int @ B @ A )
% 0.18/0.45 => ( ( ord_less_eq_int @ A @ B )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.antisym
% 0.18/0.45 thf(fact_233_dual__order_Ostrict__implies__not__eq,axiom,
% 0.18/0.45 ! [B: real,A: real] :
% 0.18/0.45 ( ( ord_less_real @ B @ A )
% 0.18/0.45 => ( A != B ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.strict_implies_not_eq
% 0.18/0.45 thf(fact_234_dual__order_Ostrict__implies__not__eq,axiom,
% 0.18/0.45 ! [B: nat,A: nat] :
% 0.18/0.45 ( ( ord_less_nat @ B @ A )
% 0.18/0.45 => ( A != B ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.strict_implies_not_eq
% 0.18/0.45 thf(fact_235_order_Ostrict__implies__not__eq,axiom,
% 0.18/0.45 ! [A: real,B: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( A != B ) ) ).
% 0.18/0.45
% 0.18/0.45 % order.strict_implies_not_eq
% 0.18/0.45 thf(fact_236_order_Ostrict__implies__not__eq,axiom,
% 0.18/0.45 ! [A: nat,B: nat] :
% 0.18/0.45 ( ( ord_less_nat @ A @ B )
% 0.18/0.45 => ( A != B ) ) ).
% 0.18/0.45
% 0.18/0.45 % order.strict_implies_not_eq
% 0.18/0.45 thf(fact_237_not__less__iff__gr__or__eq,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 0.18/0.45 = ( ( ord_less_real @ Y3 @ X )
% 0.18/0.45 | ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % not_less_iff_gr_or_eq
% 0.18/0.45 thf(fact_238_not__less__iff__gr__or__eq,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 0.18/0.45 = ( ( ord_less_nat @ Y3 @ X )
% 0.18/0.45 | ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % not_less_iff_gr_or_eq
% 0.18/0.45 thf(fact_239_dual__order_Ostrict__trans,axiom,
% 0.18/0.45 ! [B: real,A: real,C: real] :
% 0.18/0.45 ( ( ord_less_real @ B @ A )
% 0.18/0.45 => ( ( ord_less_real @ C @ B )
% 0.18/0.45 => ( ord_less_real @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.strict_trans
% 0.18/0.45 thf(fact_240_dual__order_Ostrict__trans,axiom,
% 0.18/0.45 ! [B: nat,A: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_nat @ B @ A )
% 0.18/0.45 => ( ( ord_less_nat @ C @ B )
% 0.18/0.45 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.strict_trans
% 0.18/0.45 thf(fact_241_linorder__less__wlog,axiom,
% 0.18/0.45 ! [P3: real > real > $o,A: real,B: real] :
% 0.18/0.45 ( ! [A5: real,B4: real] :
% 0.18/0.45 ( ( ord_less_real @ A5 @ B4 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( ! [A5: real] : ( P3 @ A5 @ A5 )
% 0.18/0.45 => ( ! [A5: real,B4: real] :
% 0.18/0.45 ( ( P3 @ B4 @ A5 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( P3 @ A @ B ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % linorder_less_wlog
% 0.18/0.45 thf(fact_242_linorder__less__wlog,axiom,
% 0.18/0.45 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 0.18/0.45 ( ! [A5: nat,B4: nat] :
% 0.18/0.45 ( ( ord_less_nat @ A5 @ B4 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( ! [A5: nat] : ( P3 @ A5 @ A5 )
% 0.18/0.45 => ( ! [A5: nat,B4: nat] :
% 0.18/0.45 ( ( P3 @ B4 @ A5 )
% 0.18/0.45 => ( P3 @ A5 @ B4 ) )
% 0.18/0.45 => ( P3 @ A @ B ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % linorder_less_wlog
% 0.18/0.45 thf(fact_243_exists__least__iff,axiom,
% 0.18/0.45 ( ( ^ [P5: nat > $o] :
% 0.18/0.45 ? [X6: nat] : ( P5 @ X6 ) )
% 0.18/0.45 = ( ^ [P6: nat > $o] :
% 0.18/0.45 ? [N2: nat] :
% 0.18/0.45 ( ( P6 @ N2 )
% 0.18/0.45 & ! [M2: nat] :
% 0.18/0.45 ( ( ord_less_nat @ M2 @ N2 )
% 0.18/0.45 => ~ ( P6 @ M2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % exists_least_iff
% 0.18/0.45 thf(fact_244_less__imp__not__less,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_not_less
% 0.18/0.45 thf(fact_245_less__imp__not__less,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_not_less
% 0.18/0.45 thf(fact_246_order_Ostrict__trans,axiom,
% 0.18/0.45 ! [A: real,B: real,C: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_real @ B @ C )
% 0.18/0.45 => ( ord_less_real @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order.strict_trans
% 0.18/0.45 thf(fact_247_order_Ostrict__trans,axiom,
% 0.18/0.45 ! [A: nat,B: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_nat @ A @ B )
% 0.18/0.45 => ( ( ord_less_nat @ B @ C )
% 0.18/0.45 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order.strict_trans
% 0.18/0.45 thf(fact_248_dual__order_Oirrefl,axiom,
% 0.18/0.45 ! [A: real] :
% 0.18/0.45 ~ ( ord_less_real @ A @ A ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.irrefl
% 0.18/0.45 thf(fact_249_dual__order_Oirrefl,axiom,
% 0.18/0.45 ! [A: nat] :
% 0.18/0.45 ~ ( ord_less_nat @ A @ A ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.irrefl
% 0.18/0.45 thf(fact_250_linorder__cases,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ~ ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ( ( X != Y3 )
% 0.18/0.45 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % linorder_cases
% 0.18/0.45 thf(fact_251_linorder__cases,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ~ ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ( ( X != Y3 )
% 0.18/0.45 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % linorder_cases
% 0.18/0.45 thf(fact_252_less__imp__triv,axiom,
% 0.18/0.45 ! [X: real,Y3: real,P3: $o] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_real @ Y3 @ X )
% 0.18/0.45 => P3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_triv
% 0.18/0.45 thf(fact_253_less__imp__triv,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat,P3: $o] :
% 0.18/0.45 ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_nat @ Y3 @ X )
% 0.18/0.45 => P3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_triv
% 0.18/0.45 thf(fact_254_less__imp__not__eq2,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ( Y3 != X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_not_eq2
% 0.18/0.45 thf(fact_255_less__imp__not__eq2,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ( Y3 != X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_not_eq2
% 0.18/0.45 thf(fact_256_antisym__conv3,axiom,
% 0.18/0.45 ! [Y3: real,X: real] :
% 0.18/0.45 ( ~ ( ord_less_real @ Y3 @ X )
% 0.18/0.45 => ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 0.18/0.45 = ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % antisym_conv3
% 0.18/0.45 thf(fact_257_antisym__conv3,axiom,
% 0.18/0.45 ! [Y3: nat,X: nat] :
% 0.18/0.45 ( ~ ( ord_less_nat @ Y3 @ X )
% 0.18/0.45 => ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 0.18/0.45 = ( X = Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % antisym_conv3
% 0.18/0.45 thf(fact_258_less__induct,axiom,
% 0.18/0.45 ! [P3: nat > $o,A: nat] :
% 0.18/0.45 ( ! [X2: nat] :
% 0.18/0.45 ( ! [Y5: nat] :
% 0.18/0.45 ( ( ord_less_nat @ Y5 @ X2 )
% 0.18/0.45 => ( P3 @ Y5 ) )
% 0.18/0.45 => ( P3 @ X2 ) )
% 0.18/0.45 => ( P3 @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_induct
% 0.18/0.45 thf(fact_259_less__not__sym,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_not_sym
% 0.18/0.45 thf(fact_260_less__not__sym,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_not_sym
% 0.18/0.45 thf(fact_261_less__imp__not__eq,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ( X != Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_not_eq
% 0.18/0.45 thf(fact_262_less__imp__not__eq,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ( X != Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_not_eq
% 0.18/0.45 thf(fact_263_dual__order_Oasym,axiom,
% 0.18/0.45 ! [B: real,A: real] :
% 0.18/0.45 ( ( ord_less_real @ B @ A )
% 0.18/0.45 => ~ ( ord_less_real @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.asym
% 0.18/0.45 thf(fact_264_dual__order_Oasym,axiom,
% 0.18/0.45 ! [B: nat,A: nat] :
% 0.18/0.45 ( ( ord_less_nat @ B @ A )
% 0.18/0.45 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % dual_order.asym
% 0.18/0.45 thf(fact_265_ord__less__eq__trans,axiom,
% 0.18/0.45 ! [A: real,B: real,C: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( ( B = C )
% 0.18/0.45 => ( ord_less_real @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_less_eq_trans
% 0.18/0.45 thf(fact_266_ord__less__eq__trans,axiom,
% 0.18/0.45 ! [A: nat,B: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_nat @ A @ B )
% 0.18/0.45 => ( ( B = C )
% 0.18/0.45 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_less_eq_trans
% 0.18/0.45 thf(fact_267_ord__eq__less__trans,axiom,
% 0.18/0.45 ! [A: real,B: real,C: real] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 => ( ( ord_less_real @ B @ C )
% 0.18/0.45 => ( ord_less_real @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_less_trans
% 0.18/0.45 thf(fact_268_ord__eq__less__trans,axiom,
% 0.18/0.45 ! [A: nat,B: nat,C: nat] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 => ( ( ord_less_nat @ B @ C )
% 0.18/0.45 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_less_trans
% 0.18/0.45 thf(fact_269_less__irrefl,axiom,
% 0.18/0.45 ! [X: real] :
% 0.18/0.45 ~ ( ord_less_real @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % less_irrefl
% 0.18/0.45 thf(fact_270_less__irrefl,axiom,
% 0.18/0.45 ! [X: nat] :
% 0.18/0.45 ~ ( ord_less_nat @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % less_irrefl
% 0.18/0.45 thf(fact_271_less__linear,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 | ( X = Y3 )
% 0.18/0.45 | ( ord_less_real @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_linear
% 0.18/0.45 thf(fact_272_less__linear,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 | ( X = Y3 )
% 0.18/0.45 | ( ord_less_nat @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_linear
% 0.18/0.45 thf(fact_273_less__trans,axiom,
% 0.18/0.45 ! [X: real,Y3: real,Z5: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_real @ Y3 @ Z5 )
% 0.18/0.45 => ( ord_less_real @ X @ Z5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_trans
% 0.18/0.45 thf(fact_274_less__trans,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat,Z5: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ( ( ord_less_nat @ Y3 @ Z5 )
% 0.18/0.45 => ( ord_less_nat @ X @ Z5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_trans
% 0.18/0.45 thf(fact_275_less__asym_H,axiom,
% 0.18/0.45 ! [A: real,B: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_asym'
% 0.18/0.45 thf(fact_276_less__asym_H,axiom,
% 0.18/0.45 ! [A: nat,B: nat] :
% 0.18/0.45 ( ( ord_less_nat @ A @ B )
% 0.18/0.45 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_asym'
% 0.18/0.45 thf(fact_277_less__asym,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_asym
% 0.18/0.45 thf(fact_278_less__asym,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_asym
% 0.18/0.45 thf(fact_279_less__imp__neq,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ( X != Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_neq
% 0.18/0.45 thf(fact_280_less__imp__neq,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ( X != Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_imp_neq
% 0.18/0.45 thf(fact_281_dense,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ? [Z3: real] :
% 0.18/0.45 ( ( ord_less_real @ X @ Z3 )
% 0.18/0.45 & ( ord_less_real @ Z3 @ Y3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dense
% 0.18/0.45 thf(fact_282_order_Oasym,axiom,
% 0.18/0.45 ! [A: real,B: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % order.asym
% 0.18/0.45 thf(fact_283_order_Oasym,axiom,
% 0.18/0.45 ! [A: nat,B: nat] :
% 0.18/0.45 ( ( ord_less_nat @ A @ B )
% 0.18/0.45 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % order.asym
% 0.18/0.45 thf(fact_284_neq__iff,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( X != Y3 )
% 0.18/0.45 = ( ( ord_less_real @ X @ Y3 )
% 0.18/0.45 | ( ord_less_real @ Y3 @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % neq_iff
% 0.18/0.45 thf(fact_285_neq__iff,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( X != Y3 )
% 0.18/0.45 = ( ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 | ( ord_less_nat @ Y3 @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % neq_iff
% 0.18/0.45 thf(fact_286_neqE,axiom,
% 0.18/0.45 ! [X: real,Y3: real] :
% 0.18/0.45 ( ( X != Y3 )
% 0.18/0.45 => ( ~ ( ord_less_real @ X @ Y3 )
% 0.18/0.45 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % neqE
% 0.18/0.45 thf(fact_287_neqE,axiom,
% 0.18/0.45 ! [X: nat,Y3: nat] :
% 0.18/0.45 ( ( X != Y3 )
% 0.18/0.45 => ( ~ ( ord_less_nat @ X @ Y3 )
% 0.18/0.45 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % neqE
% 0.18/0.45 thf(fact_288_gt__ex,axiom,
% 0.18/0.45 ! [X: real] :
% 0.18/0.45 ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 0.18/0.45
% 0.18/0.45 % gt_ex
% 0.18/0.45 thf(fact_289_gt__ex,axiom,
% 0.18/0.45 ! [X: nat] :
% 0.18/0.45 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.18/0.45
% 0.18/0.45 % gt_ex
% 0.18/0.45 thf(fact_290_lt__ex,axiom,
% 0.18/0.45 ! [X: real] :
% 0.18/0.45 ? [Y: real] : ( ord_less_real @ Y @ X ) ).
% 0.18/0.45
% 0.18/0.45 % lt_ex
% 0.18/0.45 thf(fact_291_order__less__subst2,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > real,C: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_less_subst2
% 0.18/0.45 thf(fact_292_order__less__subst2,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_less_subst2
% 0.18/0.45 thf(fact_293_order__less__subst2,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.18/0.45 ( ( ord_less_nat @ A @ B )
% 0.18/0.45 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_less_subst2
% 0.18/0.45 thf(fact_294_order__less__subst2,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_nat @ A @ B )
% 0.18/0.45 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_less_subst2
% 0.18/0.45 thf(fact_295_order__less__subst1,axiom,
% 0.18/0.45 ! [A: real,F: real > real,B: real,C: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_less_subst1
% 0.18/0.45 thf(fact_296_order__less__subst1,axiom,
% 0.18/0.45 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_less_subst1
% 0.18/0.45 thf(fact_297_order__less__subst1,axiom,
% 0.18/0.45 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.18/0.45 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_less_subst1
% 0.18/0.45 thf(fact_298_order__less__subst1,axiom,
% 0.18/0.45 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.18/0.45 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_less_subst1
% 0.18/0.45 thf(fact_299_ord__less__eq__subst,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > real,C: real] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_less_eq_subst
% 0.18/0.45 thf(fact_300_ord__less__eq__subst,axiom,
% 0.18/0.45 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_real @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_less_eq_subst
% 0.18/0.45 thf(fact_301_ord__less__eq__subst,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.18/0.45 ( ( ord_less_nat @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_less_eq_subst
% 0.18/0.45 thf(fact_302_ord__less__eq__subst,axiom,
% 0.18/0.45 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.18/0.45 ( ( ord_less_nat @ A @ B )
% 0.18/0.45 => ( ( ( F @ B )
% 0.18/0.45 = C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_less_eq_subst
% 0.18/0.45 thf(fact_303_ord__eq__less__subst,axiom,
% 0.18/0.45 ! [A: real,F: real > real,B: real,C: real] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_less_subst
% 0.18/0.45 thf(fact_304_ord__eq__less__subst,axiom,
% 0.18/0.45 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_real @ B @ C )
% 0.18/0.45 => ( ! [X2: real,Y: real] :
% 0.18/0.45 ( ( ord_less_real @ X2 @ Y )
% 0.18/0.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_less_subst
% 0.18/0.45 thf(fact_305_ord__eq__less__subst,axiom,
% 0.18/0.45 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_less_subst
% 0.18/0.45 thf(fact_306_ord__eq__less__subst,axiom,
% 0.18/0.45 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( F @ B ) )
% 0.18/0.45 => ( ( ord_less_nat @ B @ C )
% 0.18/0.45 => ( ! [X2: nat,Y: nat] :
% 0.18/0.45 ( ( ord_less_nat @ X2 @ Y )
% 0.18/0.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.45 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ord_eq_less_subst
% 0.18/0.45 thf(fact_307_verit__comp__simplify1_I1_J,axiom,
% 0.18/0.45 ! [A: real] :
% 0.18/0.45 ~ ( ord_less_real @ A @ A ) ).
% 0.30/0.45
% 0.30/0.45 % verit_comp_simplify1(1)
% 0.30/0.45 thf(fact_308_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.45 ! [A: nat] :
% 0.30/0.45 ~ ( ord_less_nat @ A @ A ) ).
% 0.30/0.45
% 0.30/0.45 % verit_comp_simplify1(1)
% 0.30/0.45 thf(fact_309_bot__nat__0_Oextremum__strict,axiom,
% 0.30/0.45 ! [A: nat] :
% 0.30/0.45 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.30/0.45
% 0.30/0.45 % bot_nat_0.extremum_strict
% 0.30/0.45 thf(fact_310_infinite__descent0,axiom,
% 0.30/0.45 ! [P3: nat > $o,N: nat] :
% 0.30/0.45 ( ( P3 @ zero_zero_nat )
% 0.30/0.45 => ( ! [N3: nat] :
% 0.30/0.45 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.45 => ( ~ ( P3 @ N3 )
% 0.30/0.45 => ? [M3: nat] :
% 0.30/0.45 ( ( ord_less_nat @ M3 @ N3 )
% 0.30/0.45 & ~ ( P3 @ M3 ) ) ) )
% 0.30/0.45 => ( P3 @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % infinite_descent0
% 0.30/0.45 thf(fact_311_gr__implies__not0,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_nat @ M @ N )
% 0.30/0.45 => ( N != zero_zero_nat ) ) ).
% 0.30/0.45
% 0.30/0.45 % gr_implies_not0
% 0.30/0.45 thf(fact_312_less__zeroE,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.45
% 0.30/0.45 % less_zeroE
% 0.30/0.45 thf(fact_313_not__less0,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.45
% 0.30/0.45 % not_less0
% 0.30/0.45 thf(fact_314_not__gr0,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.30/0.45 = ( N = zero_zero_nat ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_gr0
% 0.30/0.45 thf(fact_315_gr0I,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ( ( N != zero_zero_nat )
% 0.30/0.45 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % gr0I
% 0.30/0.45 thf(fact_316_cross__def,axiom,
% 0.30/0.45 ( sturm_1953858694_cross
% 0.30/0.45 = ( ^ [P2: poly_real,A2: real,B3: real] : ( sturm_1279781401iation @ ( poly_real2 @ P2 @ A2 ) @ ( poly_real2 @ P2 @ B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % cross_def
% 0.30/0.45 thf(fact_317_nat_Oinject,axiom,
% 0.30/0.45 ! [X22: nat,Y22: nat] :
% 0.30/0.45 ( ( ( suc @ X22 )
% 0.30/0.45 = ( suc @ Y22 ) )
% 0.30/0.45 = ( X22 = Y22 ) ) ).
% 0.30/0.45
% 0.30/0.45 % nat.inject
% 0.30/0.45 thf(fact_318_old_Onat_Oinject,axiom,
% 0.30/0.45 ! [Nat: nat,Nat2: nat] :
% 0.30/0.45 ( ( ( suc @ Nat )
% 0.30/0.45 = ( suc @ Nat2 ) )
% 0.30/0.45 = ( Nat = Nat2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.nat.inject
% 0.30/0.45 thf(fact_319_lessI,axiom,
% 0.30/0.45 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % lessI
% 0.30/0.45 thf(fact_320_Suc__mono,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_nat @ M @ N )
% 0.30/0.45 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Suc_mono
% 0.30/0.45 thf(fact_321_Suc__less__eq,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.45 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % Suc_less_eq
% 0.30/0.45 thf(fact_322_Suc__le__mono,axiom,
% 0.30/0.45 ! [N: nat,M: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.30/0.45 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.45
% 0.30/0.45 % Suc_le_mono
% 0.30/0.45 thf(fact_323_less__Suc0,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.45 = ( N = zero_zero_nat ) ) ).
% 0.30/0.45
% 0.30/0.45 % less_Suc0
% 0.30/0.45 thf(fact_324_zero__less__Suc,axiom,
% 0.30/0.45 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % zero_less_Suc
% 0.30/0.45 thf(fact_325_less__one,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ( ( ord_less_nat @ N @ one_one_nat )
% 0.30/0.45 = ( N = zero_zero_nat ) ) ).
% 0.30/0.45
% 0.30/0.45 % less_one
% 0.30/0.45 thf(fact_326_variation__0_I2_J,axiom,
% 0.30/0.45 ! [X: real] :
% 0.30/0.45 ( ( sturm_1279781401iation @ X @ zero_zero_real )
% 0.30/0.45 = zero_zero_int ) ).
% 0.30/0.45
% 0.30/0.45 % variation_0(2)
% 0.30/0.45 thf(fact_327_variation__0_I1_J,axiom,
% 0.30/0.45 ! [Y3: real] :
% 0.30/0.45 ( ( sturm_1279781401iation @ zero_zero_real @ Y3 )
% 0.30/0.45 = zero_zero_int ) ).
% 0.30/0.45
% 0.30/0.45 % variation_0(1)
% 0.30/0.45 thf(fact_328_One__nat__def,axiom,
% 0.30/0.45 ( one_one_nat
% 0.30/0.45 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.45
% 0.30/0.45 % One_nat_def
% 0.30/0.45 thf(fact_329_transitive__stepwise__le,axiom,
% 0.30/0.45 ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.30/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.45 => ( ! [X2: nat] : ( R @ X2 @ X2 )
% 0.30/0.45 => ( ! [X2: nat,Y: nat,Z3: nat] :
% 0.30/0.45 ( ( R @ X2 @ Y )
% 0.30/0.45 => ( ( R @ Y @ Z3 )
% 0.30/0.45 => ( R @ X2 @ Z3 ) ) )
% 0.30/0.45 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 0.30/0.45 => ( R @ M @ N ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % transitive_stepwise_le
% 0.30/0.45 thf(fact_330_nat__induct__at__least,axiom,
% 0.30/0.45 ! [M: nat,N: nat,P3: nat > $o] :
% 0.30/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.45 => ( ( P3 @ M )
% 0.30/0.45 => ( ! [N3: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ M @ N3 )
% 0.30/0.45 => ( ( P3 @ N3 )
% 0.30/0.45 => ( P3 @ ( suc @ N3 ) ) ) )
% 0.30/0.45 => ( P3 @ N ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % nat_induct_at_least
% 0.30/0.45 thf(fact_331_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.45 ! [P3: nat > $o,K2: nat,B: nat] :
% 0.30/0.45 ( ( P3 @ K2 )
% 0.30/0.45 => ( ! [Y: nat] :
% 0.30/0.45 ( ( P3 @ Y )
% 0.30/0.45 => ( ord_less_eq_nat @ Y @ B ) )
% 0.30/0.45 => ? [X2: nat] :
% 0.30/0.45 ( ( P3 @ X2 )
% 0.30/0.45 & ! [Y5: nat] :
% 0.30/0.45 ( ( P3 @ Y5 )
% 0.30/0.45 => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Nat.ex_has_greatest_nat
% 0.30/0.45 thf(fact_332_verit__la__generic,axiom,
% 0.30/0.45 ! [A: int,X: int] :
% 0.30/0.45 ( ( ord_less_eq_int @ A @ X )
% 0.30/0.45 | ( A = X )
% 0.30/0.45 | ( ord_less_eq_int @ X @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 % verit_la_generic
% 0.30/0.45 thf(fact_333_full__nat__induct,axiom,
% 0.30/0.45 ! [P3: nat > $o,N: nat] :
% 0.30/0.45 ( ! [N3: nat] :
% 0.30/0.45 ( ! [M3: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
% 0.30/0.45 => ( P3 @ M3 ) )
% 0.30/0.45 => ( P3 @ N3 ) )
% 0.30/0.45 => ( P3 @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % full_nat_induct
% 0.30/0.45 thf(fact_334_not__less__eq__eq,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.30/0.45 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_less_eq_eq
% 0.30/0.45 thf(fact_335_Suc__n__not__le__n,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.30/0.45
% 0.30/0.45 % Suc_n_not_le_n
% 0.30/0.45 thf(fact_336_nat__le__linear,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.45 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.45
% 0.30/0.45 % nat_le_linear
% 0.30/0.45 thf(fact_337_le__antisym,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.45 => ( M = N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_antisym
% 0.30/0.45 thf(fact_338_le__Suc__eq,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.45 = ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.45 | ( M
% 0.30/0.45 = ( suc @ N ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_Suc_eq
% 0.30/0.45 thf(fact_339_eq__imp__le,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( M = N )
% 0.30/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % eq_imp_le
% 0.30/0.45 thf(fact_340_le__trans,axiom,
% 0.30/0.45 ! [I: nat,J: nat,K2: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.45 => ( ( ord_less_eq_nat @ J @ K2 )
% 0.30/0.45 => ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_trans
% 0.30/0.45 thf(fact_341_Suc__le__D,axiom,
% 0.30/0.45 ! [N: nat,M4: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
% 0.30/0.45 => ? [M5: nat] :
% 0.30/0.45 ( M4
% 0.30/0.45 = ( suc @ M5 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Suc_le_D
% 0.30/0.45 thf(fact_342_le__refl,axiom,
% 0.30/0.45 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.30/0.45
% 0.30/0.45 % le_refl
% 0.30/0.45 thf(fact_343_le__SucI,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.45 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_SucI
% 0.30/0.45 thf(fact_344_le__SucE,axiom,
% 0.30/0.45 ! [M: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.45 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.45 => ( M
% 0.30/0.45 = ( suc @ N ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_SucE
% 0.30/0.45
% 0.30/0.45 % Conjectures (1)
% 0.30/0.45 thf(conj_0,conjecture,
% 0.30/0.45 ( ( sturm_469514713_r_pos @ p @ x )
% 0.30/0.45 != ( ~ ( ( ( ( poly_real2 @ p @ x )
% 0.30/0.45 = zero_zero_real )
% 0.30/0.45 => ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x ) )
% 0.30/0.45 & ( ( ( poly_real2 @ p @ x )
% 0.30/0.45 != zero_zero_real )
% 0.30/0.45 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 ------------------------
% 3.05/1.53 ---- Embedded in THF ---
% 3.05/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.05/1.53 %%% Generated on Tue Feb 24 08:29:19 EST 2026
% 3.05/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 3.05/1.53 %%% Logic specification used:
% 3.05/1.53 %%% thf(spec, logic, $$dhol).
% 3.05/1.53
% 3.05/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.JHQKkcEisk/DTF2DTF31714.p
% See solution above
% 3.05/1.53 ------------------------
% 3.05/1.57 ---- Cleaned THF ---
% 3.05/1.58 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 3.05/1.58 poly_poly_poly_real: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.05/1.58 poly_poly_real: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 3.05/1.58 poly_poly_nat: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
% 3.05/1.58 poly_poly_int: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 poly_real: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 3.05/1.58 poly_nat: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.05/1.58 poly_int: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 3.05/1.58 set_real: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Real__Oreal,type,
% 3.05/1.58 real: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Nat__Onat,type,
% 3.05/1.58 nat: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(ty_n_t__Int__Oint,type,
% 3.05/1.58 int: $tType ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 3.05/1.58 one_one_nat: nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 3.05/1.58 zero_zero_int: int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 3.05/1.58 zero_zero_nat: nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.05/1.58 zero_zero_poly_int: poly_int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 3.05/1.58 zero_zero_poly_nat: poly_nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
% 3.05/1.58 zero_z1549157189ly_int: poly_poly_int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 3.05/1.58 zero_z1059985641ly_nat: poly_poly_nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 3.05/1.58 zero_z935034829y_real: poly_poly_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.05/1.58 zero_z1423781445y_real: poly_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 zero_zero_poly_real: poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 3.05/1.58 zero_zero_real: real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Nat_OSuc,type,
% 3.05/1.58 suc: nat > nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 3.05/1.58 ord_less_int: int > int > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 3.05/1.58 ord_less_nat: nat > nat > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.05/1.58 ord_less_poly_int: poly_int > poly_int > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.05/1.58 ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 ord_less_poly_real: poly_real > poly_real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 3.05/1.58 ord_less_real: real > real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 3.05/1.58 ord_less_eq_int: int > int > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 3.05/1.58 ord_less_eq_nat: nat > nat > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.05/1.58 ord_less_eq_poly_int: poly_int > poly_int > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.05/1.58 ord_le893774876y_real: poly_poly_real > poly_poly_real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 ord_le1180086932y_real: poly_real > poly_real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 3.05/1.58 ord_less_eq_real: real > real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Ois__zero_001t__Int__Oint,type,
% 3.05/1.58 is_zero_int: poly_int > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat,type,
% 3.05/1.58 is_zero_nat: poly_nat > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 is_zero_poly_real: poly_poly_real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Ois__zero_001t__Real__Oreal,type,
% 3.05/1.58 is_zero_real: poly_real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oorder_001t__Int__Oint,type,
% 3.05/1.58 order_int: int > poly_int > nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.05/1.58 order_poly_int: poly_int > poly_poly_int > nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.05/1.58 order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 order_poly_real: poly_real > poly_poly_real > nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oorder_001t__Real__Oreal,type,
% 3.05/1.58 order_real: real > poly_real > nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opderiv_001t__Int__Oint,type,
% 3.05/1.58 pderiv_int: poly_int > poly_int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opderiv_001t__Nat__Onat,type,
% 3.05/1.58 pderiv_nat: poly_nat > poly_nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opderiv_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 pderiv_poly_real: poly_poly_real > poly_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opderiv_001t__Real__Oreal,type,
% 3.05/1.58 pderiv_real: poly_real > poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly_001t__Int__Oint,type,
% 3.05/1.58 poly_int2: poly_int > int > int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
% 3.05/1.58 poly_nat2: poly_nat > nat > nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.05/1.58 poly_poly_int2: poly_poly_int > poly_int > poly_int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 3.05/1.58 poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.05/1.58 poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 poly_poly_real2: poly_poly_real > poly_real > poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
% 3.05/1.58 poly_real2: poly_real > real > real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly__cutoff_001t__Int__Oint,type,
% 3.05/1.58 poly_cutoff_int: nat > poly_int > poly_int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat,type,
% 3.05/1.58 poly_cutoff_nat: nat > poly_nat > poly_nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly__cutoff_001t__Real__Oreal,type,
% 3.05/1.58 poly_cutoff_real: nat > poly_real > poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly__shift_001t__Int__Oint,type,
% 3.05/1.58 poly_shift_int: nat > poly_int > poly_int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat,type,
% 3.05/1.58 poly_shift_nat: nat > poly_nat > poly_nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Opoly__shift_001t__Real__Oreal,type,
% 3.05/1.58 poly_shift_real: nat > poly_real > poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oreflect__poly_001t__Int__Oint,type,
% 3.05/1.58 reflect_poly_int: poly_int > poly_int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
% 3.05/1.58 reflect_poly_nat: poly_nat > poly_nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.05/1.58 reflec943087998ly_int: poly_poly_int > poly_poly_int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 3.05/1.58 reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.05/1.58 reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.05/1.58 reflec1522834046y_real: poly_poly_real > poly_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal,type,
% 3.05/1.58 reflect_poly_real: poly_real > poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Polynomial_Orsquarefree_001t__Real__Oreal,type,
% 3.05/1.58 rsquarefree_real: poly_real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 3.05/1.58 collect_real: ( real > $o ) > set_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Ocross,type,
% 3.05/1.58 sturm_1953858694_cross: poly_real > real > real > int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Osign__r__pos,type,
% 3.05/1.58 sturm_469514713_r_pos: poly_real > real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Ovariation,type,
% 3.05/1.58 sturm_1279781401iation: real > real > int ).
% 3.05/1.58
% 3.05/1.58 thf(sy_c_member_001t__Real__Oreal,type,
% 3.05/1.58 member_real: real > set_real > $o ).
% 3.05/1.58
% 3.05/1.58 thf(sy_v_p,type,
% 3.05/1.58 p: poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(sy_v_x,type,
% 3.05/1.58 x: real ).
% 3.05/1.58
% 3.05/1.58 thf(fact_0_False,axiom,
% 3.05/1.58 ( ( poly_real2 @ p @ x )
% 3.05/1.58 != zero_zero_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_1__092_060open_062sign__r__pos_Ap_Ax_A_092_060Longrightarrow_062_A0_A_060_Apoly_Ap_Ax_092_060close_062,axiom,
% 3.05/1.58 ( ( sturm_469514713_r_pos @ p @ x )
% 3.05/1.58 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_2__092_060open_0620_A_060_Apoly_Ap_Ax_A_092_060Longrightarrow_062_Asign__r__pos_Ap_Ax_092_060close_062,axiom,
% 3.05/1.58 ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) )
% 3.05/1.58 => ( sturm_469514713_r_pos @ p @ x ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_3_assms,axiom,
% 3.05/1.58 p != zero_zero_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(fact_4_poly__0,axiom,
% 3.05/1.58 ! [X: poly_nat] :
% 3.05/1.58 ( ( poly_poly_nat2 @ zero_z1059985641ly_nat @ X )
% 3.05/1.58 = zero_zero_poly_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_5_poly__0,axiom,
% 3.05/1.58 ! [X: poly_int] :
% 3.05/1.58 ( ( poly_poly_int2 @ zero_z1549157189ly_int @ X )
% 3.05/1.58 = zero_zero_poly_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_6_poly__0,axiom,
% 3.05/1.58 ! [X: poly_poly_real] :
% 3.05/1.58 ( ( poly_poly_poly_real2 @ zero_z935034829y_real @ X )
% 3.05/1.58 = zero_z1423781445y_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_7_poly__0,axiom,
% 3.05/1.58 ! [X: poly_real] :
% 3.05/1.58 ( ( poly_poly_real2 @ zero_z1423781445y_real @ X )
% 3.05/1.58 = zero_zero_poly_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_8_poly__0,axiom,
% 3.05/1.58 ! [X: int] :
% 3.05/1.58 ( ( poly_int2 @ zero_zero_poly_int @ X )
% 3.05/1.58 = zero_zero_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_9_poly__0,axiom,
% 3.05/1.58 ! [X: nat] :
% 3.05/1.58 ( ( poly_nat2 @ zero_zero_poly_nat @ X )
% 3.05/1.58 = zero_zero_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_10_poly__0,axiom,
% 3.05/1.58 ! [X: real] :
% 3.05/1.58 ( ( poly_real2 @ zero_zero_poly_real @ X )
% 3.05/1.58 = zero_zero_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_11_not__gr__zero,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 3.05/1.58 = ( N = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_12_poly__IVT__neg,axiom,
% 3.05/1.58 ! [A: real,B: real,P: poly_real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A ) )
% 3.05/1.58 => ( ( ord_less_real @ ( poly_real2 @ P @ B ) @ zero_zero_real )
% 3.05/1.58 => ? [X2: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ X2 )
% 3.05/1.58 & ( ord_less_real @ X2 @ B )
% 3.05/1.58 & ( ( poly_real2 @ P @ X2 )
% 3.05/1.58 = zero_zero_real ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_13_poly__IVT__pos,axiom,
% 3.05/1.58 ! [A: real,B: real,P: poly_real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_real @ ( poly_real2 @ P @ A ) @ zero_zero_real )
% 3.05/1.58 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B ) )
% 3.05/1.58 => ? [X2: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ X2 )
% 3.05/1.58 & ( ord_less_real @ X2 @ B )
% 3.05/1.58 & ( ( poly_real2 @ P @ X2 )
% 3.05/1.58 = zero_zero_real ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_14_poly__all__0__iff__0,axiom,
% 3.05/1.58 ! [P: poly_poly_int] :
% 3.05/1.58 ( ( ! [X3: poly_int] :
% 3.05/1.58 ( ( poly_poly_int2 @ P @ X3 )
% 3.05/1.58 = zero_zero_poly_int ) )
% 3.05/1.58 = ( P = zero_z1549157189ly_int ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_15_poly__all__0__iff__0,axiom,
% 3.05/1.58 ! [P: poly_poly_poly_real] :
% 3.05/1.58 ( ( ! [X3: poly_poly_real] :
% 3.05/1.58 ( ( poly_poly_poly_real2 @ P @ X3 )
% 3.05/1.58 = zero_z1423781445y_real ) )
% 3.05/1.58 = ( P = zero_z935034829y_real ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_16_poly__all__0__iff__0,axiom,
% 3.05/1.58 ! [P: poly_real] :
% 3.05/1.58 ( ( ! [X3: real] :
% 3.05/1.58 ( ( poly_real2 @ P @ X3 )
% 3.05/1.58 = zero_zero_real ) )
% 3.05/1.58 = ( P = zero_zero_poly_real ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_17_poly__all__0__iff__0,axiom,
% 3.05/1.58 ! [P: poly_poly_real] :
% 3.05/1.58 ( ( ! [X3: poly_real] :
% 3.05/1.58 ( ( poly_poly_real2 @ P @ X3 )
% 3.05/1.58 = zero_zero_poly_real ) )
% 3.05/1.58 = ( P = zero_z1423781445y_real ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_18_poly__all__0__iff__0,axiom,
% 3.05/1.58 ! [P: poly_int] :
% 3.05/1.58 ( ( ! [X3: int] :
% 3.05/1.58 ( ( poly_int2 @ P @ X3 )
% 3.05/1.58 = zero_zero_int ) )
% 3.05/1.58 = ( P = zero_zero_poly_int ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_19_less__numeral__extra_I3_J,axiom,
% 3.05/1.58 ~ ( ord_less_poly_real @ zero_zero_poly_real @ zero_zero_poly_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_20_less__numeral__extra_I3_J,axiom,
% 3.05/1.58 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_21_less__numeral__extra_I3_J,axiom,
% 3.05/1.58 ~ ( ord_less_poly_int @ zero_zero_poly_int @ zero_zero_poly_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_22_less__numeral__extra_I3_J,axiom,
% 3.05/1.58 ~ ( ord_le38482960y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_23_less__numeral__extra_I3_J,axiom,
% 3.05/1.58 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_24_less__numeral__extra_I3_J,axiom,
% 3.05/1.58 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_25_field__lbound__gt__zero,axiom,
% 3.05/1.58 ! [D1: real,D2: real] :
% 3.05/1.58 ( ( ord_less_real @ zero_zero_real @ D1 )
% 3.05/1.58 => ( ( ord_less_real @ zero_zero_real @ D2 )
% 3.05/1.58 => ? [E: real] :
% 3.05/1.58 ( ( ord_less_real @ zero_zero_real @ E )
% 3.05/1.58 & ( ord_less_real @ E @ D1 )
% 3.05/1.58 & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_26_gr__zeroI,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( N != zero_zero_nat )
% 3.05/1.58 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_27_not__less__zero,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_28_gr__implies__not__zero,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_nat @ M @ N )
% 3.05/1.58 => ( N != zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_29_zero__less__iff__neq__zero,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.05/1.58 = ( N != zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_30_pderiv__0,axiom,
% 3.05/1.58 ( ( pderiv_real @ zero_zero_poly_real )
% 3.05/1.58 = zero_zero_poly_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_31_pderiv__0,axiom,
% 3.05/1.58 ( ( pderiv_nat @ zero_zero_poly_nat )
% 3.05/1.58 = zero_zero_poly_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_32_pderiv__0,axiom,
% 3.05/1.58 ( ( pderiv_int @ zero_zero_poly_int )
% 3.05/1.58 = zero_zero_poly_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_33_pderiv__0,axiom,
% 3.05/1.58 ( ( pderiv_poly_real @ zero_z1423781445y_real )
% 3.05/1.58 = zero_z1423781445y_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_34_zero__reorient,axiom,
% 3.05/1.58 ! [X: real] :
% 3.05/1.58 ( ( zero_zero_real = X )
% 3.05/1.58 = ( X = zero_zero_real ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_35_zero__reorient,axiom,
% 3.05/1.58 ! [X: poly_real] :
% 3.05/1.58 ( ( zero_zero_poly_real = X )
% 3.05/1.58 = ( X = zero_zero_poly_real ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_36_zero__reorient,axiom,
% 3.05/1.58 ! [X: int] :
% 3.05/1.58 ( ( zero_zero_int = X )
% 3.05/1.58 = ( X = zero_zero_int ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_37_zero__reorient,axiom,
% 3.05/1.58 ! [X: nat] :
% 3.05/1.58 ( ( zero_zero_nat = X )
% 3.05/1.58 = ( X = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_38_zero__reorient,axiom,
% 3.05/1.58 ! [X: poly_nat] :
% 3.05/1.58 ( ( zero_zero_poly_nat = X )
% 3.05/1.58 = ( X = zero_zero_poly_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_39_zero__reorient,axiom,
% 3.05/1.58 ! [X: poly_int] :
% 3.05/1.58 ( ( zero_zero_poly_int = X )
% 3.05/1.58 = ( X = zero_zero_poly_int ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_40_zero__reorient,axiom,
% 3.05/1.58 ! [X: poly_poly_real] :
% 3.05/1.58 ( ( zero_z1423781445y_real = X )
% 3.05/1.58 = ( X = zero_z1423781445y_real ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_41_poly__eq__poly__eq__iff,axiom,
% 3.05/1.58 ! [P: poly_real,Q: poly_real] :
% 3.05/1.58 ( ( ( poly_real2 @ P )
% 3.05/1.58 = ( poly_real2 @ Q ) )
% 3.05/1.58 = ( P = Q ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_42_poly__eq__poly__eq__iff,axiom,
% 3.05/1.58 ! [P: poly_int,Q: poly_int] :
% 3.05/1.58 ( ( ( poly_int2 @ P )
% 3.05/1.58 = ( poly_int2 @ Q ) )
% 3.05/1.58 = ( P = Q ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_43_poly__eq__poly__eq__iff,axiom,
% 3.05/1.58 ! [P: poly_poly_real,Q: poly_poly_real] :
% 3.05/1.58 ( ( ( poly_poly_real2 @ P )
% 3.05/1.58 = ( poly_poly_real2 @ Q ) )
% 3.05/1.58 = ( P = Q ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_44_cross__0,axiom,
% 3.05/1.58 ! [A: real,B: real] :
% 3.05/1.58 ( ( sturm_1953858694_cross @ zero_zero_poly_real @ A @ B )
% 3.05/1.58 = zero_zero_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_45_rsquarefree__roots,axiom,
% 3.05/1.58 ( rsquarefree_real
% 3.05/1.58 = ( ^ [P2: poly_real] :
% 3.05/1.58 ! [A2: real] :
% 3.05/1.58 ~ ( ( ( poly_real2 @ P2 @ A2 )
% 3.05/1.58 = zero_zero_real )
% 3.05/1.58 & ( ( poly_real2 @ ( pderiv_real @ P2 ) @ A2 )
% 3.05/1.58 = zero_zero_real ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_46_cross__no__root,axiom,
% 3.05/1.58 ! [A: real,B: real,P: poly_real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( ! [X2: real] :
% 3.05/1.58 ( ( ( ord_less_real @ A @ X2 )
% 3.05/1.58 & ( ord_less_real @ X2 @ B ) )
% 3.05/1.58 => ( ( poly_real2 @ P @ X2 )
% 3.05/1.58 != zero_zero_real ) )
% 3.05/1.58 => ( ( sturm_1953858694_cross @ P @ A @ B )
% 3.05/1.58 = zero_zero_int ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_47_is__zero__null,axiom,
% 3.05/1.58 ( is_zero_real
% 3.05/1.58 = ( ^ [P2: poly_real] : ( P2 = zero_zero_poly_real ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_48_is__zero__null,axiom,
% 3.05/1.58 ( is_zero_nat
% 3.05/1.58 = ( ^ [P2: poly_nat] : ( P2 = zero_zero_poly_nat ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_49_is__zero__null,axiom,
% 3.05/1.58 ( is_zero_int
% 3.05/1.58 = ( ^ [P2: poly_int] : ( P2 = zero_zero_poly_int ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_50_is__zero__null,axiom,
% 3.05/1.58 ( is_zero_poly_real
% 3.05/1.58 = ( ^ [P2: poly_poly_real] : ( P2 = zero_z1423781445y_real ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_51_poly__cutoff__0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( poly_cutoff_real @ N @ zero_zero_poly_real )
% 3.05/1.58 = zero_zero_poly_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_52_poly__cutoff__0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
% 3.05/1.58 = zero_zero_poly_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_53_poly__cutoff__0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( poly_cutoff_int @ N @ zero_zero_poly_int )
% 3.05/1.58 = zero_zero_poly_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_54_poly__cutoff__0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( poly_c1404107022y_real @ N @ zero_z1423781445y_real )
% 3.05/1.58 = zero_z1423781445y_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_55_reflect__poly__at__0__eq__0__iff,axiom,
% 3.05/1.58 ! [P: poly_real] :
% 3.05/1.58 ( ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
% 3.05/1.58 = zero_zero_real )
% 3.05/1.58 = ( P = zero_zero_poly_real ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_56_reflect__poly__at__0__eq__0__iff,axiom,
% 3.05/1.58 ! [P: poly_poly_real] :
% 3.05/1.58 ( ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
% 3.05/1.58 = zero_zero_poly_real )
% 3.05/1.58 = ( P = zero_z1423781445y_real ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_57_reflect__poly__at__0__eq__0__iff,axiom,
% 3.05/1.58 ! [P: poly_int] :
% 3.05/1.58 ( ( ( poly_int2 @ ( reflect_poly_int @ P ) @ zero_zero_int )
% 3.05/1.58 = zero_zero_int )
% 3.05/1.58 = ( P = zero_zero_poly_int ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_58_reflect__poly__at__0__eq__0__iff,axiom,
% 3.05/1.58 ! [P: poly_nat] :
% 3.05/1.58 ( ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 3.05/1.58 = zero_zero_nat )
% 3.05/1.58 = ( P = zero_zero_poly_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_59_reflect__poly__at__0__eq__0__iff,axiom,
% 3.05/1.58 ! [P: poly_poly_nat] :
% 3.05/1.58 ( ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
% 3.05/1.58 = zero_zero_poly_nat )
% 3.05/1.58 = ( P = zero_z1059985641ly_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_60_reflect__poly__at__0__eq__0__iff,axiom,
% 3.05/1.58 ! [P: poly_poly_int] :
% 3.05/1.58 ( ( ( poly_poly_int2 @ ( reflec943087998ly_int @ P ) @ zero_zero_poly_int )
% 3.05/1.58 = zero_zero_poly_int )
% 3.05/1.58 = ( P = zero_z1549157189ly_int ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_61_reflect__poly__at__0__eq__0__iff,axiom,
% 3.05/1.58 ! [P: poly_poly_poly_real] :
% 3.05/1.58 ( ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
% 3.05/1.58 = zero_z1423781445y_real )
% 3.05/1.58 = ( P = zero_z935034829y_real ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_62_next__non__root__interval,axiom,
% 3.05/1.58 ! [P: poly_real,Lb: real] :
% 3.05/1.58 ( ( P != zero_zero_poly_real )
% 3.05/1.58 => ~ ! [Ub: real] :
% 3.05/1.58 ( ( ord_less_real @ Lb @ Ub )
% 3.05/1.58 => ~ ! [Z: real] :
% 3.05/1.58 ( ( ( ord_less_real @ Lb @ Z )
% 3.05/1.58 & ( ord_less_eq_real @ Z @ Ub ) )
% 3.05/1.58 => ( ( poly_real2 @ P @ Z )
% 3.05/1.58 != zero_zero_real ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_63_last__non__root__interval,axiom,
% 3.05/1.58 ! [P: poly_real,Ub2: real] :
% 3.05/1.58 ( ( P != zero_zero_poly_real )
% 3.05/1.58 => ~ ! [Lb2: real] :
% 3.05/1.58 ( ( ord_less_real @ Lb2 @ Ub2 )
% 3.05/1.58 => ~ ! [Z: real] :
% 3.05/1.58 ( ( ( ord_less_eq_real @ Lb2 @ Z )
% 3.05/1.58 & ( ord_less_real @ Z @ Ub2 ) )
% 3.05/1.58 => ( ( poly_real2 @ P @ Z )
% 3.05/1.58 != zero_zero_real ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_64_poly__shift__0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( poly_shift_real @ N @ zero_zero_poly_real )
% 3.05/1.58 = zero_zero_poly_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_65_poly__shift__0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( poly_shift_nat @ N @ zero_zero_poly_nat )
% 3.05/1.58 = zero_zero_poly_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_66_poly__shift__0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( poly_shift_int @ N @ zero_zero_poly_int )
% 3.05/1.58 = zero_zero_poly_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_67_poly__shift__0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( poly_shift_poly_real @ N @ zero_z1423781445y_real )
% 3.05/1.58 = zero_z1423781445y_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_68_order__root,axiom,
% 3.05/1.58 ! [P: poly_real,A: real] :
% 3.05/1.58 ( ( ( poly_real2 @ P @ A )
% 3.05/1.58 = zero_zero_real )
% 3.05/1.58 = ( ( P = zero_zero_poly_real )
% 3.05/1.58 | ( ( order_real @ A @ P )
% 3.05/1.58 != zero_zero_nat ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_69_order__root,axiom,
% 3.05/1.58 ! [P: poly_poly_real,A: poly_real] :
% 3.05/1.58 ( ( ( poly_poly_real2 @ P @ A )
% 3.05/1.58 = zero_zero_poly_real )
% 3.05/1.58 = ( ( P = zero_z1423781445y_real )
% 3.05/1.58 | ( ( order_poly_real @ A @ P )
% 3.05/1.58 != zero_zero_nat ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_70_order__root,axiom,
% 3.05/1.58 ! [P: poly_int,A: int] :
% 3.05/1.58 ( ( ( poly_int2 @ P @ A )
% 3.05/1.58 = zero_zero_int )
% 3.05/1.58 = ( ( P = zero_zero_poly_int )
% 3.05/1.58 | ( ( order_int @ A @ P )
% 3.05/1.58 != zero_zero_nat ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_71_order__root,axiom,
% 3.05/1.58 ! [P: poly_poly_int,A: poly_int] :
% 3.05/1.58 ( ( ( poly_poly_int2 @ P @ A )
% 3.05/1.58 = zero_zero_poly_int )
% 3.05/1.58 = ( ( P = zero_z1549157189ly_int )
% 3.05/1.58 | ( ( order_poly_int @ A @ P )
% 3.05/1.58 != zero_zero_nat ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_72_order__root,axiom,
% 3.05/1.58 ! [P: poly_poly_poly_real,A: poly_poly_real] :
% 3.05/1.58 ( ( ( poly_poly_poly_real2 @ P @ A )
% 3.05/1.58 = zero_z1423781445y_real )
% 3.05/1.58 = ( ( P = zero_z935034829y_real )
% 3.05/1.58 | ( ( order_poly_poly_real @ A @ P )
% 3.05/1.58 != zero_zero_nat ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_73_le__zero__eq,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 3.05/1.58 = ( N = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_74_reflect__poly__0,axiom,
% 3.05/1.58 ( ( reflect_poly_real @ zero_zero_poly_real )
% 3.05/1.58 = zero_zero_poly_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_75_reflect__poly__0,axiom,
% 3.05/1.58 ( ( reflect_poly_nat @ zero_zero_poly_nat )
% 3.05/1.58 = zero_zero_poly_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_76_reflect__poly__0,axiom,
% 3.05/1.58 ( ( reflect_poly_int @ zero_zero_poly_int )
% 3.05/1.58 = zero_zero_poly_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_77_reflect__poly__0,axiom,
% 3.05/1.58 ( ( reflec1522834046y_real @ zero_z1423781445y_real )
% 3.05/1.58 = zero_z1423781445y_real ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_78_complete__real,axiom,
% 3.05/1.58 ! [S: set_real] :
% 3.05/1.58 ( ? [X4: real] : ( member_real @ X4 @ S )
% 3.05/1.58 => ( ? [Z: real] :
% 3.05/1.58 ! [X2: real] :
% 3.05/1.58 ( ( member_real @ X2 @ S )
% 3.05/1.58 => ( ord_less_eq_real @ X2 @ Z ) )
% 3.05/1.58 => ? [Y: real] :
% 3.05/1.58 ( ! [X4: real] :
% 3.05/1.58 ( ( member_real @ X4 @ S )
% 3.05/1.58 => ( ord_less_eq_real @ X4 @ Y ) )
% 3.05/1.58 & ! [Z: real] :
% 3.05/1.58 ( ! [X2: real] :
% 3.05/1.58 ( ( member_real @ X2 @ S )
% 3.05/1.58 => ( ord_less_eq_real @ X2 @ Z ) )
% 3.05/1.58 => ( ord_less_eq_real @ Y @ Z ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_79_le__numeral__extra_I3_J,axiom,
% 3.05/1.58 ord_le1180086932y_real @ zero_zero_poly_real @ zero_zero_poly_real ).
% 3.05/1.58
% 3.05/1.58 thf(fact_80_le__numeral__extra_I3_J,axiom,
% 3.05/1.58 ord_less_eq_poly_int @ zero_zero_poly_int @ zero_zero_poly_int ).
% 3.05/1.58
% 3.05/1.58 thf(fact_81_le__numeral__extra_I3_J,axiom,
% 3.05/1.58 ord_le893774876y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ).
% 3.05/1.58
% 3.05/1.58 thf(fact_82_le__numeral__extra_I3_J,axiom,
% 3.05/1.58 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 3.05/1.58
% 3.05/1.58 thf(fact_83_le__numeral__extra_I3_J,axiom,
% 3.05/1.58 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 3.05/1.58
% 3.05/1.58 thf(fact_84_le__numeral__extra_I3_J,axiom,
% 3.05/1.58 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 3.05/1.58
% 3.05/1.58 thf(fact_85_zero__le,axiom,
% 3.05/1.58 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_86_less__eq__real__def,axiom,
% 3.05/1.58 ( ord_less_eq_real
% 3.05/1.58 = ( ^ [X3: real,Y2: real] :
% 3.05/1.58 ( ( ord_less_real @ X3 @ Y2 )
% 3.05/1.58 | ( X3 = Y2 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_87_order__0I,axiom,
% 3.05/1.58 ! [P: poly_real,A: real] :
% 3.05/1.58 ( ( ( poly_real2 @ P @ A )
% 3.05/1.58 != zero_zero_real )
% 3.05/1.58 => ( ( order_real @ A @ P )
% 3.05/1.58 = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_88_order__0I,axiom,
% 3.05/1.58 ! [P: poly_poly_real,A: poly_real] :
% 3.05/1.58 ( ( ( poly_poly_real2 @ P @ A )
% 3.05/1.58 != zero_zero_poly_real )
% 3.05/1.58 => ( ( order_poly_real @ A @ P )
% 3.05/1.58 = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_89_order__0I,axiom,
% 3.05/1.58 ! [P: poly_int,A: int] :
% 3.05/1.58 ( ( ( poly_int2 @ P @ A )
% 3.05/1.58 != zero_zero_int )
% 3.05/1.58 => ( ( order_int @ A @ P )
% 3.05/1.58 = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_90_order__0I,axiom,
% 3.05/1.58 ! [P: poly_poly_int,A: poly_int] :
% 3.05/1.58 ( ( ( poly_poly_int2 @ P @ A )
% 3.05/1.58 != zero_zero_poly_int )
% 3.05/1.58 => ( ( order_poly_int @ A @ P )
% 3.05/1.58 = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_91_order__0I,axiom,
% 3.05/1.58 ! [P: poly_poly_poly_real,A: poly_poly_real] :
% 3.05/1.58 ( ( ( poly_poly_poly_real2 @ P @ A )
% 3.05/1.58 != zero_z1423781445y_real )
% 3.05/1.58 => ( ( order_poly_poly_real @ A @ P )
% 3.05/1.58 = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_92_not__eq__pos__or__neg__iff__1,axiom,
% 3.05/1.58 ! [Lb: real,Ub2: real,P: poly_real] :
% 3.05/1.58 ( ( ! [Z2: real] :
% 3.05/1.58 ( ( ( ord_less_real @ Lb @ Z2 )
% 3.05/1.58 & ( ord_less_eq_real @ Z2 @ Ub2 ) )
% 3.05/1.58 => ( ( poly_real2 @ P @ Z2 )
% 3.05/1.58 != zero_zero_real ) ) )
% 3.05/1.58 = ( ! [Z2: real] :
% 3.05/1.58 ( ( ( ord_less_real @ Lb @ Z2 )
% 3.05/1.58 & ( ord_less_eq_real @ Z2 @ Ub2 ) )
% 3.05/1.58 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z2 ) ) )
% 3.05/1.58 | ! [Z2: real] :
% 3.05/1.58 ( ( ( ord_less_real @ Lb @ Z2 )
% 3.05/1.58 & ( ord_less_eq_real @ Z2 @ Ub2 ) )
% 3.05/1.58 => ( ord_less_real @ ( poly_real2 @ P @ Z2 ) @ zero_zero_real ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_93_not__eq__pos__or__neg__iff__2,axiom,
% 3.05/1.58 ! [Lb: real,Ub2: real,P: poly_real] :
% 3.05/1.58 ( ( ! [Z2: real] :
% 3.05/1.58 ( ( ( ord_less_eq_real @ Lb @ Z2 )
% 3.05/1.58 & ( ord_less_real @ Z2 @ Ub2 ) )
% 3.05/1.58 => ( ( poly_real2 @ P @ Z2 )
% 3.05/1.58 != zero_zero_real ) ) )
% 3.05/1.58 = ( ! [Z2: real] :
% 3.05/1.58 ( ( ( ord_less_eq_real @ Lb @ Z2 )
% 3.05/1.58 & ( ord_less_real @ Z2 @ Ub2 ) )
% 3.05/1.58 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z2 ) ) )
% 3.05/1.58 | ! [Z2: real] :
% 3.05/1.58 ( ( ( ord_less_eq_real @ Lb @ Z2 )
% 3.05/1.58 & ( ord_less_real @ Z2 @ Ub2 ) )
% 3.05/1.58 => ( ord_less_real @ ( poly_real2 @ P @ Z2 ) @ zero_zero_real ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_94_neq0__conv,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( N != zero_zero_nat )
% 3.05/1.58 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_95_less__nat__zero__code,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_96_bot__nat__0_Onot__eq__extremum,axiom,
% 3.05/1.58 ! [A: nat] :
% 3.05/1.58 ( ( A != zero_zero_nat )
% 3.05/1.58 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_97_order__refl,axiom,
% 3.05/1.58 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_98_order__refl,axiom,
% 3.05/1.58 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_99_order__refl,axiom,
% 3.05/1.58 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_100_minf_I8_J,axiom,
% 3.05/1.58 ! [T: real] :
% 3.05/1.58 ? [Z3: real] :
% 3.05/1.58 ! [X4: real] :
% 3.05/1.58 ( ( ord_less_real @ X4 @ Z3 )
% 3.05/1.58 => ~ ( ord_less_eq_real @ T @ X4 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_101_minf_I8_J,axiom,
% 3.05/1.58 ! [T: nat] :
% 3.05/1.58 ? [Z3: nat] :
% 3.05/1.58 ! [X4: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X4 @ Z3 )
% 3.05/1.58 => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_102_minf_I8_J,axiom,
% 3.05/1.58 ! [T: int] :
% 3.05/1.58 ? [Z3: int] :
% 3.05/1.58 ! [X4: int] :
% 3.05/1.58 ( ( ord_less_int @ X4 @ Z3 )
% 3.05/1.58 => ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_103_minf_I6_J,axiom,
% 3.05/1.58 ! [T: real] :
% 3.05/1.58 ? [Z3: real] :
% 3.05/1.58 ! [X4: real] :
% 3.05/1.58 ( ( ord_less_real @ X4 @ Z3 )
% 3.05/1.58 => ( ord_less_eq_real @ X4 @ T ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_104_minf_I6_J,axiom,
% 3.05/1.58 ! [T: nat] :
% 3.05/1.58 ? [Z3: nat] :
% 3.05/1.58 ! [X4: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X4 @ Z3 )
% 3.05/1.58 => ( ord_less_eq_nat @ X4 @ T ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_105_minf_I6_J,axiom,
% 3.05/1.58 ! [T: int] :
% 3.05/1.58 ? [Z3: int] :
% 3.05/1.58 ! [X4: int] :
% 3.05/1.58 ( ( ord_less_int @ X4 @ Z3 )
% 3.05/1.58 => ( ord_less_eq_int @ X4 @ T ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_106_mem__Collect__eq,axiom,
% 3.05/1.58 ! [A: real,P3: real > $o] :
% 3.05/1.58 ( ( member_real @ A @ ( collect_real @ P3 ) )
% 3.05/1.58 = ( P3 @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_107_Collect__mem__eq,axiom,
% 3.05/1.58 ! [A3: set_real] :
% 3.05/1.58 ( ( collect_real
% 3.05/1.58 @ ^ [X3: real] : ( member_real @ X3 @ A3 ) )
% 3.05/1.58 = A3 ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_108_pinf_I8_J,axiom,
% 3.05/1.58 ! [T: real] :
% 3.05/1.58 ? [Z3: real] :
% 3.05/1.58 ! [X4: real] :
% 3.05/1.58 ( ( ord_less_real @ Z3 @ X4 )
% 3.05/1.58 => ( ord_less_eq_real @ T @ X4 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_109_pinf_I8_J,axiom,
% 3.05/1.58 ! [T: nat] :
% 3.05/1.58 ? [Z3: nat] :
% 3.05/1.58 ! [X4: nat] :
% 3.05/1.58 ( ( ord_less_nat @ Z3 @ X4 )
% 3.05/1.58 => ( ord_less_eq_nat @ T @ X4 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_110_pinf_I8_J,axiom,
% 3.05/1.58 ! [T: int] :
% 3.05/1.58 ? [Z3: int] :
% 3.05/1.58 ! [X4: int] :
% 3.05/1.58 ( ( ord_less_int @ Z3 @ X4 )
% 3.05/1.58 => ( ord_less_eq_int @ T @ X4 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_111_pinf_I6_J,axiom,
% 3.05/1.58 ! [T: real] :
% 3.05/1.58 ? [Z3: real] :
% 3.05/1.58 ! [X4: real] :
% 3.05/1.58 ( ( ord_less_real @ Z3 @ X4 )
% 3.05/1.58 => ~ ( ord_less_eq_real @ X4 @ T ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_112_pinf_I6_J,axiom,
% 3.05/1.58 ! [T: nat] :
% 3.05/1.58 ? [Z3: nat] :
% 3.05/1.58 ! [X4: nat] :
% 3.05/1.58 ( ( ord_less_nat @ Z3 @ X4 )
% 3.05/1.58 => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_113_pinf_I6_J,axiom,
% 3.05/1.58 ! [T: int] :
% 3.05/1.58 ? [Z3: int] :
% 3.05/1.58 ! [X4: int] :
% 3.05/1.58 ( ( ord_less_int @ Z3 @ X4 )
% 3.05/1.58 => ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_114_verit__comp__simplify1_I3_J,axiom,
% 3.05/1.58 ! [B2: real,A4: real] :
% 3.05/1.58 ( ( ~ ( ord_less_eq_real @ B2 @ A4 ) )
% 3.05/1.58 = ( ord_less_real @ A4 @ B2 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_115_verit__comp__simplify1_I3_J,axiom,
% 3.05/1.58 ! [B2: nat,A4: nat] :
% 3.05/1.58 ( ( ~ ( ord_less_eq_nat @ B2 @ A4 ) )
% 3.05/1.58 = ( ord_less_nat @ A4 @ B2 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_116_verit__comp__simplify1_I3_J,axiom,
% 3.05/1.58 ! [B2: int,A4: int] :
% 3.05/1.58 ( ( ~ ( ord_less_eq_int @ B2 @ A4 ) )
% 3.05/1.58 = ( ord_less_int @ A4 @ B2 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_117_bot__nat__0_Oextremum,axiom,
% 3.05/1.58 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_118_le0,axiom,
% 3.05/1.58 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_119_nat__less__le,axiom,
% 3.05/1.58 ( ord_less_nat
% 3.05/1.58 = ( ^ [M2: nat,N2: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.05/1.58 & ( M2 != N2 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_120_less__imp__le__nat,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_nat @ M @ N )
% 3.05/1.58 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_121_le__eq__less__or__eq,axiom,
% 3.05/1.58 ( ord_less_eq_nat
% 3.05/1.58 = ( ^ [M2: nat,N2: nat] :
% 3.05/1.58 ( ( ord_less_nat @ M2 @ N2 )
% 3.05/1.58 | ( M2 = N2 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_122_less__or__eq__imp__le,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ( ord_less_nat @ M @ N )
% 3.05/1.58 | ( M = N ) )
% 3.05/1.58 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_123_le__neq__implies__less,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M @ N )
% 3.05/1.58 => ( ( M != N )
% 3.05/1.58 => ( ord_less_nat @ M @ N ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_124_less__mono__imp__le__mono,axiom,
% 3.05/1.58 ! [F: nat > nat,I: nat,J: nat] :
% 3.05/1.58 ( ! [I2: nat,J2: nat] :
% 3.05/1.58 ( ( ord_less_nat @ I2 @ J2 )
% 3.05/1.58 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 3.05/1.58 => ( ( ord_less_eq_nat @ I @ J )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_125_linorder__neqE__nat,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( X != Y3 )
% 3.05/1.58 => ( ~ ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_126_infinite__descent,axiom,
% 3.05/1.58 ! [P3: nat > $o,N: nat] :
% 3.05/1.58 ( ! [N3: nat] :
% 3.05/1.58 ( ~ ( P3 @ N3 )
% 3.05/1.58 => ? [M3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ M3 @ N3 )
% 3.05/1.58 & ~ ( P3 @ M3 ) ) )
% 3.05/1.58 => ( P3 @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_127_nat__less__induct,axiom,
% 3.05/1.58 ! [P3: nat > $o,N: nat] :
% 3.05/1.58 ( ! [N3: nat] :
% 3.05/1.58 ( ! [M3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ M3 @ N3 )
% 3.05/1.58 => ( P3 @ M3 ) )
% 3.05/1.58 => ( P3 @ N3 ) )
% 3.05/1.58 => ( P3 @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_128_less__irrefl__nat,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ N @ N ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_129_less__not__refl3,axiom,
% 3.05/1.58 ! [S2: nat,T: nat] :
% 3.05/1.58 ( ( ord_less_nat @ S2 @ T )
% 3.05/1.58 => ( S2 != T ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_130_less__not__refl2,axiom,
% 3.05/1.58 ! [N: nat,M: nat] :
% 3.05/1.58 ( ( ord_less_nat @ N @ M )
% 3.05/1.58 => ( M != N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_131_less__not__refl,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ N @ N ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_132_nat__neq__iff,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( M != N )
% 3.05/1.58 = ( ( ord_less_nat @ M @ N )
% 3.05/1.58 | ( ord_less_nat @ N @ M ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_133_bot__nat__0_Oextremum__uniqueI,axiom,
% 3.05/1.58 ! [A: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 3.05/1.58 => ( A = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_134_bot__nat__0_Oextremum__unique,axiom,
% 3.05/1.58 ! [A: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 3.05/1.58 = ( A = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_135_ex__least__nat__le,axiom,
% 3.05/1.58 ! [P3: nat > $o,N: nat] :
% 3.05/1.58 ( ( P3 @ N )
% 3.05/1.58 => ( ~ ( P3 @ zero_zero_nat )
% 3.05/1.58 => ? [K: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ K @ N )
% 3.05/1.58 & ! [I3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ I3 @ K )
% 3.05/1.58 => ~ ( P3 @ I3 ) )
% 3.05/1.58 & ( P3 @ K ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_136_le__0__eq,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 3.05/1.58 = ( N = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_137_less__eq__nat_Osimps_I1_J,axiom,
% 3.05/1.58 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_138_conj__le__cong,axiom,
% 3.05/1.58 ! [X: int,X5: int,P3: $o,P4: $o] :
% 3.05/1.58 ( ( X = X5 )
% 3.05/1.58 => ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 3.05/1.58 => ( P3 = P4 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.05/1.58 & P3 )
% 3.05/1.58 = ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 3.05/1.58 & P4 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_139_imp__le__cong,axiom,
% 3.05/1.58 ! [X: int,X5: int,P3: $o,P4: $o] :
% 3.05/1.58 ( ( X = X5 )
% 3.05/1.58 => ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 3.05/1.58 => ( P3 = P4 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.05/1.58 => P3 )
% 3.05/1.58 = ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 3.05/1.58 => P4 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_140_order__subst1,axiom,
% 3.05/1.58 ! [A: real,F: real > real,B: real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_141_order__subst1,axiom,
% 3.05/1.58 ! [A: real,F: nat > real,B: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_142_order__subst1,axiom,
% 3.05/1.58 ! [A: real,F: int > real,B: int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_int @ B @ C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_143_order__subst1,axiom,
% 3.05/1.58 ! [A: nat,F: real > nat,B: real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_144_order__subst1,axiom,
% 3.05/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_145_order__subst1,axiom,
% 3.05/1.58 ! [A: nat,F: int > nat,B: int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_int @ B @ C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_146_order__subst1,axiom,
% 3.05/1.58 ! [A: int,F: real > int,B: real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_147_order__subst1,axiom,
% 3.05/1.58 ! [A: int,F: nat > int,B: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_148_order__subst1,axiom,
% 3.05/1.58 ! [A: int,F: int > int,B: int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_int @ B @ C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_149_order__subst2,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_150_order__subst2,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_151_order__subst2,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_152_order__subst2,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_153_order__subst2,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_154_order__subst2,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_155_order__subst2,axiom,
% 3.05/1.58 ! [A: int,B: int,F: int > real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_156_order__subst2,axiom,
% 3.05/1.58 ! [A: int,B: int,F: int > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_157_order__subst2,axiom,
% 3.05/1.58 ! [A: int,B: int,F: int > int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_158_verit__la__disequality,axiom,
% 3.05/1.58 ! [A: real,B: real] :
% 3.05/1.58 ( ( A = B )
% 3.05/1.58 | ~ ( ord_less_eq_real @ A @ B )
% 3.05/1.58 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_159_verit__la__disequality,axiom,
% 3.05/1.58 ! [A: nat,B: nat] :
% 3.05/1.58 ( ( A = B )
% 3.05/1.58 | ~ ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_160_verit__la__disequality,axiom,
% 3.05/1.58 ! [A: int,B: int] :
% 3.05/1.58 ( ( A = B )
% 3.05/1.58 | ~ ( ord_less_eq_int @ A @ B )
% 3.05/1.58 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_161_ord__eq__le__subst,axiom,
% 3.05/1.58 ! [A: real,F: real > real,B: real,C: real] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_162_ord__eq__le__subst,axiom,
% 3.05/1.58 ! [A: nat,F: real > nat,B: real,C: real] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_163_ord__eq__le__subst,axiom,
% 3.05/1.58 ! [A: int,F: real > int,B: real,C: real] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_164_ord__eq__le__subst,axiom,
% 3.05/1.58 ! [A: real,F: nat > real,B: nat,C: nat] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_165_ord__eq__le__subst,axiom,
% 3.05/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_166_ord__eq__le__subst,axiom,
% 3.05/1.58 ! [A: int,F: nat > int,B: nat,C: nat] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_167_ord__eq__le__subst,axiom,
% 3.05/1.58 ! [A: real,F: int > real,B: int,C: int] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_int @ B @ C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_168_ord__eq__le__subst,axiom,
% 3.05/1.58 ! [A: nat,F: int > nat,B: int,C: int] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_int @ B @ C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_169_ord__eq__le__subst,axiom,
% 3.05/1.58 ! [A: int,F: int > int,B: int,C: int] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_eq_int @ B @ C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_170_ord__le__eq__subst,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_171_ord__le__eq__subst,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_172_ord__le__eq__subst,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_173_ord__le__eq__subst,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_174_ord__le__eq__subst,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_175_ord__le__eq__subst,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_176_ord__le__eq__subst,axiom,
% 3.05/1.58 ! [A: int,B: int,F: int > real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_177_ord__le__eq__subst,axiom,
% 3.05/1.58 ! [A: int,B: int,F: int > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_178_ord__le__eq__subst,axiom,
% 3.05/1.58 ! [A: int,B: int,F: int > int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: int,Y: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X2 @ Y )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_179_eq__iff,axiom,
% 3.05/1.58 ( ( ^ [Y4: real,Z4: real] : ( Y4 = Z4 ) )
% 3.05/1.58 = ( ^ [X3: real,Y2: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X3 @ Y2 )
% 3.05/1.58 & ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_180_eq__iff,axiom,
% 3.05/1.58 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 3.05/1.58 = ( ^ [X3: nat,Y2: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 3.05/1.58 & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_181_eq__iff,axiom,
% 3.05/1.58 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 3.05/1.58 = ( ^ [X3: int,Y2: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X3 @ Y2 )
% 3.05/1.58 & ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_182_antisym,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_eq_real @ Y3 @ X )
% 3.05/1.58 => ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_183_antisym,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_eq_nat @ Y3 @ X )
% 3.05/1.58 => ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_184_antisym,axiom,
% 3.05/1.58 ! [X: int,Y3: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_eq_int @ Y3 @ X )
% 3.05/1.58 => ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_185_linear,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X @ Y3 )
% 3.05/1.58 | ( ord_less_eq_real @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_186_linear,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X @ Y3 )
% 3.05/1.58 | ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_187_linear,axiom,
% 3.05/1.58 ! [X: int,Y3: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X @ Y3 )
% 3.05/1.58 | ( ord_less_eq_int @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_188_eq__refl,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( X = Y3 )
% 3.05/1.58 => ( ord_less_eq_real @ X @ Y3 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_189_eq__refl,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( X = Y3 )
% 3.05/1.58 => ( ord_less_eq_nat @ X @ Y3 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_190_eq__refl,axiom,
% 3.05/1.58 ! [X: int,Y3: int] :
% 3.05/1.58 ( ( X = Y3 )
% 3.05/1.58 => ( ord_less_eq_int @ X @ Y3 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_191_le__cases,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ~ ( ord_less_eq_real @ X @ Y3 )
% 3.05/1.58 => ( ord_less_eq_real @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_192_le__cases,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ~ ( ord_less_eq_nat @ X @ Y3 )
% 3.05/1.58 => ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_193_le__cases,axiom,
% 3.05/1.58 ! [X: int,Y3: int] :
% 3.05/1.58 ( ~ ( ord_less_eq_int @ X @ Y3 )
% 3.05/1.58 => ( ord_less_eq_int @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_194_order_Otrans,axiom,
% 3.05/1.58 ! [A: real,B: real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_real @ B @ C )
% 3.05/1.58 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_195_order_Otrans,axiom,
% 3.05/1.58 ! [A: nat,B: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 3.05/1.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_196_order_Otrans,axiom,
% 3.05/1.58 ! [A: int,B: int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_int @ B @ C )
% 3.05/1.58 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_197_le__cases3,axiom,
% 3.05/1.58 ! [X: real,Y3: real,Z5: real] :
% 3.05/1.58 ( ( ( ord_less_eq_real @ X @ Y3 )
% 3.05/1.58 => ~ ( ord_less_eq_real @ Y3 @ Z5 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_real @ Y3 @ X )
% 3.05/1.58 => ~ ( ord_less_eq_real @ X @ Z5 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_real @ X @ Z5 )
% 3.05/1.58 => ~ ( ord_less_eq_real @ Z5 @ Y3 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_real @ Z5 @ Y3 )
% 3.05/1.58 => ~ ( ord_less_eq_real @ Y3 @ X ) )
% 3.05/1.58 => ( ( ( ord_less_eq_real @ Y3 @ Z5 )
% 3.05/1.58 => ~ ( ord_less_eq_real @ Z5 @ X ) )
% 3.05/1.58 => ~ ( ( ord_less_eq_real @ Z5 @ X )
% 3.05/1.58 => ~ ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_198_le__cases3,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat,Z5: nat] :
% 3.05/1.58 ( ( ( ord_less_eq_nat @ X @ Y3 )
% 3.05/1.58 => ~ ( ord_less_eq_nat @ Y3 @ Z5 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_nat @ Y3 @ X )
% 3.05/1.58 => ~ ( ord_less_eq_nat @ X @ Z5 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_nat @ X @ Z5 )
% 3.05/1.58 => ~ ( ord_less_eq_nat @ Z5 @ Y3 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_nat @ Z5 @ Y3 )
% 3.05/1.58 => ~ ( ord_less_eq_nat @ Y3 @ X ) )
% 3.05/1.58 => ( ( ( ord_less_eq_nat @ Y3 @ Z5 )
% 3.05/1.58 => ~ ( ord_less_eq_nat @ Z5 @ X ) )
% 3.05/1.58 => ~ ( ( ord_less_eq_nat @ Z5 @ X )
% 3.05/1.58 => ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_199_le__cases3,axiom,
% 3.05/1.58 ! [X: int,Y3: int,Z5: int] :
% 3.05/1.58 ( ( ( ord_less_eq_int @ X @ Y3 )
% 3.05/1.58 => ~ ( ord_less_eq_int @ Y3 @ Z5 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_int @ Y3 @ X )
% 3.05/1.58 => ~ ( ord_less_eq_int @ X @ Z5 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_int @ X @ Z5 )
% 3.05/1.58 => ~ ( ord_less_eq_int @ Z5 @ Y3 ) )
% 3.05/1.58 => ( ( ( ord_less_eq_int @ Z5 @ Y3 )
% 3.05/1.58 => ~ ( ord_less_eq_int @ Y3 @ X ) )
% 3.05/1.58 => ( ( ( ord_less_eq_int @ Y3 @ Z5 )
% 3.05/1.58 => ~ ( ord_less_eq_int @ Z5 @ X ) )
% 3.05/1.58 => ~ ( ( ord_less_eq_int @ Z5 @ X )
% 3.05/1.58 => ~ ( ord_less_eq_int @ X @ Y3 ) ) ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_200_antisym__conv,axiom,
% 3.05/1.58 ! [Y3: real,X: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ Y3 @ X )
% 3.05/1.58 => ( ( ord_less_eq_real @ X @ Y3 )
% 3.05/1.58 = ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_201_antisym__conv,axiom,
% 3.05/1.58 ! [Y3: nat,X: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ Y3 @ X )
% 3.05/1.58 => ( ( ord_less_eq_nat @ X @ Y3 )
% 3.05/1.58 = ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_202_antisym__conv,axiom,
% 3.05/1.58 ! [Y3: int,X: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ Y3 @ X )
% 3.05/1.58 => ( ( ord_less_eq_int @ X @ Y3 )
% 3.05/1.58 = ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_203_order__class_Oorder_Oeq__iff,axiom,
% 3.05/1.58 ( ( ^ [Y4: real,Z4: real] : ( Y4 = Z4 ) )
% 3.05/1.58 = ( ^ [A2: real,B3: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ A2 @ B3 )
% 3.05/1.58 & ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_204_order__class_Oorder_Oeq__iff,axiom,
% 3.05/1.58 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 3.05/1.58 = ( ^ [A2: nat,B3: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.05/1.58 & ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_205_order__class_Oorder_Oeq__iff,axiom,
% 3.05/1.58 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 3.05/1.58 = ( ^ [A2: int,B3: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ A2 @ B3 )
% 3.05/1.58 & ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_206_ord__eq__le__trans,axiom,
% 3.05/1.58 ! [A: real,B: real,C: real] :
% 3.05/1.58 ( ( A = B )
% 3.05/1.58 => ( ( ord_less_eq_real @ B @ C )
% 3.05/1.58 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_207_ord__eq__le__trans,axiom,
% 3.05/1.58 ! [A: nat,B: nat,C: nat] :
% 3.05/1.58 ( ( A = B )
% 3.05/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 3.05/1.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_208_ord__eq__le__trans,axiom,
% 3.05/1.58 ! [A: int,B: int,C: int] :
% 3.05/1.58 ( ( A = B )
% 3.05/1.58 => ( ( ord_less_eq_int @ B @ C )
% 3.05/1.58 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_209_ord__le__eq__trans,axiom,
% 3.05/1.58 ! [A: real,B: real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( ( B = C )
% 3.05/1.58 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_210_ord__le__eq__trans,axiom,
% 3.05/1.58 ! [A: nat,B: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( ( B = C )
% 3.05/1.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_211_ord__le__eq__trans,axiom,
% 3.05/1.58 ! [A: int,B: int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( ( B = C )
% 3.05/1.58 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_212_order__class_Oorder_Oantisym,axiom,
% 3.05/1.58 ! [A: real,B: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_real @ B @ A )
% 3.05/1.58 => ( A = B ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_213_order__class_Oorder_Oantisym,axiom,
% 3.05/1.58 ! [A: nat,B: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_nat @ B @ A )
% 3.05/1.58 => ( A = B ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_214_order__class_Oorder_Oantisym,axiom,
% 3.05/1.58 ! [A: int,B: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( ( ord_less_eq_int @ B @ A )
% 3.05/1.58 => ( A = B ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_215_order__trans,axiom,
% 3.05/1.58 ! [X: real,Y3: real,Z5: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_eq_real @ Y3 @ Z5 )
% 3.05/1.58 => ( ord_less_eq_real @ X @ Z5 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_216_order__trans,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat,Z5: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_eq_nat @ Y3 @ Z5 )
% 3.05/1.58 => ( ord_less_eq_nat @ X @ Z5 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_217_order__trans,axiom,
% 3.05/1.58 ! [X: int,Y3: int,Z5: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_eq_int @ Y3 @ Z5 )
% 3.05/1.58 => ( ord_less_eq_int @ X @ Z5 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_218_dual__order_Orefl,axiom,
% 3.05/1.58 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_219_dual__order_Orefl,axiom,
% 3.05/1.58 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_220_dual__order_Orefl,axiom,
% 3.05/1.58 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_221_linorder__wlog,axiom,
% 3.05/1.58 ! [P3: real > real > $o,A: real,B: real] :
% 3.05/1.58 ( ! [A5: real,B4: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ A5 @ B4 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( ! [A5: real,B4: real] :
% 3.05/1.58 ( ( P3 @ B4 @ A5 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( P3 @ A @ B ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_222_linorder__wlog,axiom,
% 3.05/1.58 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 3.05/1.58 ( ! [A5: nat,B4: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ A5 @ B4 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( ! [A5: nat,B4: nat] :
% 3.05/1.58 ( ( P3 @ B4 @ A5 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( P3 @ A @ B ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_223_linorder__wlog,axiom,
% 3.05/1.58 ! [P3: int > int > $o,A: int,B: int] :
% 3.05/1.58 ( ! [A5: int,B4: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ A5 @ B4 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( ! [A5: int,B4: int] :
% 3.05/1.58 ( ( P3 @ B4 @ A5 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( P3 @ A @ B ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_224_dual__order_Otrans,axiom,
% 3.05/1.58 ! [B: real,A: real,C: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ B @ A )
% 3.05/1.58 => ( ( ord_less_eq_real @ C @ B )
% 3.05/1.58 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_225_dual__order_Otrans,axiom,
% 3.05/1.58 ! [B: nat,A: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ B @ A )
% 3.05/1.58 => ( ( ord_less_eq_nat @ C @ B )
% 3.05/1.58 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_226_dual__order_Otrans,axiom,
% 3.05/1.58 ! [B: int,A: int,C: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ B @ A )
% 3.05/1.58 => ( ( ord_less_eq_int @ C @ B )
% 3.05/1.58 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_227_dual__order_Oeq__iff,axiom,
% 3.05/1.58 ( ( ^ [Y4: real,Z4: real] : ( Y4 = Z4 ) )
% 3.05/1.58 = ( ^ [A2: real,B3: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ B3 @ A2 )
% 3.05/1.58 & ( ord_less_eq_real @ A2 @ B3 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_228_dual__order_Oeq__iff,axiom,
% 3.05/1.58 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 3.05/1.58 = ( ^ [A2: nat,B3: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ B3 @ A2 )
% 3.05/1.58 & ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_229_dual__order_Oeq__iff,axiom,
% 3.05/1.58 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 3.05/1.58 = ( ^ [A2: int,B3: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ B3 @ A2 )
% 3.05/1.58 & ( ord_less_eq_int @ A2 @ B3 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_230_dual__order_Oantisym,axiom,
% 3.05/1.58 ! [B: real,A: real] :
% 3.05/1.58 ( ( ord_less_eq_real @ B @ A )
% 3.05/1.58 => ( ( ord_less_eq_real @ A @ B )
% 3.05/1.58 => ( A = B ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_231_dual__order_Oantisym,axiom,
% 3.05/1.58 ! [B: nat,A: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ B @ A )
% 3.05/1.58 => ( ( ord_less_eq_nat @ A @ B )
% 3.05/1.58 => ( A = B ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_232_dual__order_Oantisym,axiom,
% 3.05/1.58 ! [B: int,A: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ B @ A )
% 3.05/1.58 => ( ( ord_less_eq_int @ A @ B )
% 3.05/1.58 => ( A = B ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_233_dual__order_Ostrict__implies__not__eq,axiom,
% 3.05/1.58 ! [B: real,A: real] :
% 3.05/1.58 ( ( ord_less_real @ B @ A )
% 3.05/1.58 => ( A != B ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_234_dual__order_Ostrict__implies__not__eq,axiom,
% 3.05/1.58 ! [B: nat,A: nat] :
% 3.05/1.58 ( ( ord_less_nat @ B @ A )
% 3.05/1.58 => ( A != B ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_235_order_Ostrict__implies__not__eq,axiom,
% 3.05/1.58 ! [A: real,B: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( A != B ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_236_order_Ostrict__implies__not__eq,axiom,
% 3.05/1.58 ! [A: nat,B: nat] :
% 3.05/1.58 ( ( ord_less_nat @ A @ B )
% 3.05/1.58 => ( A != B ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_237_not__less__iff__gr__or__eq,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 3.05/1.58 = ( ( ord_less_real @ Y3 @ X )
% 3.05/1.58 | ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_238_not__less__iff__gr__or__eq,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 3.05/1.58 = ( ( ord_less_nat @ Y3 @ X )
% 3.05/1.58 | ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_239_dual__order_Ostrict__trans,axiom,
% 3.05/1.58 ! [B: real,A: real,C: real] :
% 3.05/1.58 ( ( ord_less_real @ B @ A )
% 3.05/1.58 => ( ( ord_less_real @ C @ B )
% 3.05/1.58 => ( ord_less_real @ C @ A ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_240_dual__order_Ostrict__trans,axiom,
% 3.05/1.58 ! [B: nat,A: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_nat @ B @ A )
% 3.05/1.58 => ( ( ord_less_nat @ C @ B )
% 3.05/1.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_241_linorder__less__wlog,axiom,
% 3.05/1.58 ! [P3: real > real > $o,A: real,B: real] :
% 3.05/1.58 ( ! [A5: real,B4: real] :
% 3.05/1.58 ( ( ord_less_real @ A5 @ B4 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( ! [A5: real] : ( P3 @ A5 @ A5 )
% 3.05/1.58 => ( ! [A5: real,B4: real] :
% 3.05/1.58 ( ( P3 @ B4 @ A5 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( P3 @ A @ B ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_242_linorder__less__wlog,axiom,
% 3.05/1.58 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 3.05/1.58 ( ! [A5: nat,B4: nat] :
% 3.05/1.58 ( ( ord_less_nat @ A5 @ B4 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( ! [A5: nat] : ( P3 @ A5 @ A5 )
% 3.05/1.58 => ( ! [A5: nat,B4: nat] :
% 3.05/1.58 ( ( P3 @ B4 @ A5 )
% 3.05/1.58 => ( P3 @ A5 @ B4 ) )
% 3.05/1.58 => ( P3 @ A @ B ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_243_exists__least__iff,axiom,
% 3.05/1.58 ( ( ^ [P5: nat > $o] :
% 3.05/1.58 ? [X6: nat] : ( P5 @ X6 ) )
% 3.05/1.58 = ( ^ [P6: nat > $o] :
% 3.05/1.58 ? [N2: nat] :
% 3.05/1.58 ( ( P6 @ N2 )
% 3.05/1.58 & ! [M2: nat] :
% 3.05/1.58 ( ( ord_less_nat @ M2 @ N2 )
% 3.05/1.58 => ~ ( P6 @ M2 ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_244_less__imp__not__less,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_245_less__imp__not__less,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_246_order_Ostrict__trans,axiom,
% 3.05/1.58 ! [A: real,B: real,C: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_real @ B @ C )
% 3.05/1.58 => ( ord_less_real @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_247_order_Ostrict__trans,axiom,
% 3.05/1.58 ! [A: nat,B: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_nat @ A @ B )
% 3.05/1.58 => ( ( ord_less_nat @ B @ C )
% 3.05/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_248_dual__order_Oirrefl,axiom,
% 3.05/1.58 ! [A: real] :
% 3.05/1.58 ~ ( ord_less_real @ A @ A ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_249_dual__order_Oirrefl,axiom,
% 3.05/1.58 ! [A: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ A @ A ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_250_linorder__cases,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ~ ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ( ( X != Y3 )
% 3.05/1.58 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_251_linorder__cases,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ~ ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ( ( X != Y3 )
% 3.05/1.58 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_252_less__imp__triv,axiom,
% 3.05/1.58 ! [X: real,Y3: real,P3: $o] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_real @ Y3 @ X )
% 3.05/1.58 => P3 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_253_less__imp__triv,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat,P3: $o] :
% 3.05/1.58 ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_nat @ Y3 @ X )
% 3.05/1.58 => P3 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_254_less__imp__not__eq2,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ( Y3 != X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_255_less__imp__not__eq2,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ( Y3 != X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_256_antisym__conv3,axiom,
% 3.05/1.58 ! [Y3: real,X: real] :
% 3.05/1.58 ( ~ ( ord_less_real @ Y3 @ X )
% 3.05/1.58 => ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 3.05/1.58 = ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_257_antisym__conv3,axiom,
% 3.05/1.58 ! [Y3: nat,X: nat] :
% 3.05/1.58 ( ~ ( ord_less_nat @ Y3 @ X )
% 3.05/1.58 => ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 3.05/1.58 = ( X = Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_258_less__induct,axiom,
% 3.05/1.58 ! [P3: nat > $o,A: nat] :
% 3.05/1.58 ( ! [X2: nat] :
% 3.05/1.58 ( ! [Y5: nat] :
% 3.05/1.58 ( ( ord_less_nat @ Y5 @ X2 )
% 3.05/1.58 => ( P3 @ Y5 ) )
% 3.05/1.58 => ( P3 @ X2 ) )
% 3.05/1.58 => ( P3 @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_259_less__not__sym,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_260_less__not__sym,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_261_less__imp__not__eq,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ( X != Y3 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_262_less__imp__not__eq,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ( X != Y3 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_263_dual__order_Oasym,axiom,
% 3.05/1.58 ! [B: real,A: real] :
% 3.05/1.58 ( ( ord_less_real @ B @ A )
% 3.05/1.58 => ~ ( ord_less_real @ A @ B ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_264_dual__order_Oasym,axiom,
% 3.05/1.58 ! [B: nat,A: nat] :
% 3.05/1.58 ( ( ord_less_nat @ B @ A )
% 3.05/1.58 => ~ ( ord_less_nat @ A @ B ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_265_ord__less__eq__trans,axiom,
% 3.05/1.58 ! [A: real,B: real,C: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( ( B = C )
% 3.05/1.58 => ( ord_less_real @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_266_ord__less__eq__trans,axiom,
% 3.05/1.58 ! [A: nat,B: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_nat @ A @ B )
% 3.05/1.58 => ( ( B = C )
% 3.05/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_267_ord__eq__less__trans,axiom,
% 3.05/1.58 ! [A: real,B: real,C: real] :
% 3.05/1.58 ( ( A = B )
% 3.05/1.58 => ( ( ord_less_real @ B @ C )
% 3.05/1.58 => ( ord_less_real @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_268_ord__eq__less__trans,axiom,
% 3.05/1.58 ! [A: nat,B: nat,C: nat] :
% 3.05/1.58 ( ( A = B )
% 3.05/1.58 => ( ( ord_less_nat @ B @ C )
% 3.05/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_269_less__irrefl,axiom,
% 3.05/1.58 ! [X: real] :
% 3.05/1.58 ~ ( ord_less_real @ X @ X ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_270_less__irrefl,axiom,
% 3.05/1.58 ! [X: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ X @ X ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_271_less__linear,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 | ( X = Y3 )
% 3.05/1.58 | ( ord_less_real @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_272_less__linear,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 | ( X = Y3 )
% 3.05/1.58 | ( ord_less_nat @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_273_less__trans,axiom,
% 3.05/1.58 ! [X: real,Y3: real,Z5: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_real @ Y3 @ Z5 )
% 3.05/1.58 => ( ord_less_real @ X @ Z5 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_274_less__trans,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat,Z5: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ( ( ord_less_nat @ Y3 @ Z5 )
% 3.05/1.58 => ( ord_less_nat @ X @ Z5 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_275_less__asym_H,axiom,
% 3.05/1.58 ! [A: real,B: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ~ ( ord_less_real @ B @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_276_less__asym_H,axiom,
% 3.05/1.58 ! [A: nat,B: nat] :
% 3.05/1.58 ( ( ord_less_nat @ A @ B )
% 3.05/1.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_277_less__asym,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_278_less__asym,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_279_less__imp__neq,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ( X != Y3 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_280_less__imp__neq,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ( X != Y3 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_281_dense,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ? [Z3: real] :
% 3.05/1.58 ( ( ord_less_real @ X @ Z3 )
% 3.05/1.58 & ( ord_less_real @ Z3 @ Y3 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_282_order_Oasym,axiom,
% 3.05/1.58 ! [A: real,B: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ~ ( ord_less_real @ B @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_283_order_Oasym,axiom,
% 3.05/1.58 ! [A: nat,B: nat] :
% 3.05/1.58 ( ( ord_less_nat @ A @ B )
% 3.05/1.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_284_neq__iff,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( X != Y3 )
% 3.05/1.58 = ( ( ord_less_real @ X @ Y3 )
% 3.05/1.58 | ( ord_less_real @ Y3 @ X ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_285_neq__iff,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( X != Y3 )
% 3.05/1.58 = ( ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 | ( ord_less_nat @ Y3 @ X ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_286_neqE,axiom,
% 3.05/1.58 ! [X: real,Y3: real] :
% 3.05/1.58 ( ( X != Y3 )
% 3.05/1.58 => ( ~ ( ord_less_real @ X @ Y3 )
% 3.05/1.58 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_287_neqE,axiom,
% 3.05/1.58 ! [X: nat,Y3: nat] :
% 3.05/1.58 ( ( X != Y3 )
% 3.05/1.58 => ( ~ ( ord_less_nat @ X @ Y3 )
% 3.05/1.58 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_288_gt__ex,axiom,
% 3.05/1.58 ! [X: real] :
% 3.05/1.58 ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_289_gt__ex,axiom,
% 3.05/1.58 ! [X: nat] :
% 3.05/1.58 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_290_lt__ex,axiom,
% 3.05/1.58 ! [X: real] :
% 3.05/1.58 ? [Y: real] : ( ord_less_real @ Y @ X ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_291_order__less__subst2,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > real,C: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_real @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_292_order__less__subst2,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_293_order__less__subst2,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > real,C: real] :
% 3.05/1.58 ( ( ord_less_nat @ A @ B )
% 3.05/1.58 => ( ( ord_less_real @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_294_order__less__subst2,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_nat @ A @ B )
% 3.05/1.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_295_order__less__subst1,axiom,
% 3.05/1.58 ! [A: real,F: real > real,B: real,C: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_296_order__less__subst1,axiom,
% 3.05/1.58 ! [A: real,F: nat > real,B: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_297_order__less__subst1,axiom,
% 3.05/1.58 ! [A: nat,F: real > nat,B: real,C: real] :
% 3.05/1.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_298_order__less__subst1,axiom,
% 3.05/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 3.05/1.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_299_ord__less__eq__subst,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > real,C: real] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_300_ord__less__eq__subst,axiom,
% 3.05/1.58 ! [A: real,B: real,F: real > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_real @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_301_ord__less__eq__subst,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > real,C: real] :
% 3.05/1.58 ( ( ord_less_nat @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_302_ord__less__eq__subst,axiom,
% 3.05/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 3.05/1.58 ( ( ord_less_nat @ A @ B )
% 3.05/1.58 => ( ( ( F @ B )
% 3.05/1.58 = C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_303_ord__eq__less__subst,axiom,
% 3.05/1.58 ! [A: real,F: real > real,B: real,C: real] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_304_ord__eq__less__subst,axiom,
% 3.05/1.58 ! [A: nat,F: real > nat,B: real,C: real] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_real @ B @ C )
% 3.05/1.58 => ( ! [X2: real,Y: real] :
% 3.05/1.58 ( ( ord_less_real @ X2 @ Y )
% 3.05/1.58 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_305_ord__eq__less__subst,axiom,
% 3.05/1.58 ! [A: real,F: nat > real,B: nat,C: nat] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_306_ord__eq__less__subst,axiom,
% 3.05/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 3.05/1.58 ( ( A
% 3.05/1.58 = ( F @ B ) )
% 3.05/1.58 => ( ( ord_less_nat @ B @ C )
% 3.05/1.58 => ( ! [X2: nat,Y: nat] :
% 3.05/1.58 ( ( ord_less_nat @ X2 @ Y )
% 3.05/1.58 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.05/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_307_verit__comp__simplify1_I1_J,axiom,
% 3.05/1.58 ! [A: real] :
% 3.05/1.58 ~ ( ord_less_real @ A @ A ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_308_verit__comp__simplify1_I1_J,axiom,
% 3.05/1.58 ! [A: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ A @ A ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_309_bot__nat__0_Oextremum__strict,axiom,
% 3.05/1.58 ! [A: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_310_infinite__descent0,axiom,
% 3.05/1.58 ! [P3: nat > $o,N: nat] :
% 3.05/1.58 ( ( P3 @ zero_zero_nat )
% 3.05/1.58 => ( ! [N3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 3.05/1.58 => ( ~ ( P3 @ N3 )
% 3.05/1.58 => ? [M3: nat] :
% 3.05/1.58 ( ( ord_less_nat @ M3 @ N3 )
% 3.05/1.58 & ~ ( P3 @ M3 ) ) ) )
% 3.05/1.58 => ( P3 @ N ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_311_gr__implies__not0,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_nat @ M @ N )
% 3.05/1.58 => ( N != zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_312_less__zeroE,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_313_not__less0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_314_not__gr0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 3.05/1.58 = ( N = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_315_gr0I,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( N != zero_zero_nat )
% 3.05/1.58 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_316_cross__def,axiom,
% 3.05/1.58 ( sturm_1953858694_cross
% 3.05/1.58 = ( ^ [P2: poly_real,A2: real,B3: real] : ( sturm_1279781401iation @ ( poly_real2 @ P2 @ A2 ) @ ( poly_real2 @ P2 @ B3 ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_317_nat_Oinject,axiom,
% 3.05/1.58 ! [X22: nat,Y22: nat] :
% 3.05/1.58 ( ( ( suc @ X22 )
% 3.05/1.58 = ( suc @ Y22 ) )
% 3.05/1.58 = ( X22 = Y22 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_318_old_Onat_Oinject,axiom,
% 3.05/1.58 ! [Nat: nat,Nat2: nat] :
% 3.05/1.58 ( ( ( suc @ Nat )
% 3.05/1.58 = ( suc @ Nat2 ) )
% 3.05/1.58 = ( Nat = Nat2 ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_319_lessI,axiom,
% 3.05/1.58 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_320_Suc__mono,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_nat @ M @ N )
% 3.05/1.58 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_321_Suc__less__eq,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.05/1.58 = ( ord_less_nat @ M @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_322_Suc__le__mono,axiom,
% 3.05/1.58 ! [N: nat,M: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 3.05/1.58 = ( ord_less_eq_nat @ N @ M ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_323_less__Suc0,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 3.05/1.58 = ( N = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_324_zero__less__Suc,axiom,
% 3.05/1.58 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_325_less__one,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ( ( ord_less_nat @ N @ one_one_nat )
% 3.05/1.58 = ( N = zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_326_variation__0_I2_J,axiom,
% 3.05/1.58 ! [X: real] :
% 3.05/1.58 ( ( sturm_1279781401iation @ X @ zero_zero_real )
% 3.05/1.58 = zero_zero_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_327_variation__0_I1_J,axiom,
% 3.05/1.58 ! [Y3: real] :
% 3.05/1.58 ( ( sturm_1279781401iation @ zero_zero_real @ Y3 )
% 3.05/1.58 = zero_zero_int ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_328_One__nat__def,axiom,
% 3.05/1.58 ( one_one_nat
% 3.05/1.58 = ( suc @ zero_zero_nat ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_329_transitive__stepwise__le,axiom,
% 3.05/1.58 ! [M: nat,N: nat,R: nat > nat > $o] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M @ N )
% 3.05/1.58 => ( ! [X2: nat] : ( R @ X2 @ X2 )
% 3.05/1.58 => ( ! [X2: nat,Y: nat,Z3: nat] :
% 3.05/1.58 ( ( R @ X2 @ Y )
% 3.05/1.58 => ( ( R @ Y @ Z3 )
% 3.05/1.58 => ( R @ X2 @ Z3 ) ) )
% 3.05/1.58 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 3.05/1.58 => ( R @ M @ N ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_330_nat__induct__at__least,axiom,
% 3.05/1.58 ! [M: nat,N: nat,P3: nat > $o] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M @ N )
% 3.05/1.58 => ( ( P3 @ M )
% 3.05/1.58 => ( ! [N3: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M @ N3 )
% 3.05/1.58 => ( ( P3 @ N3 )
% 3.05/1.58 => ( P3 @ ( suc @ N3 ) ) ) )
% 3.05/1.58 => ( P3 @ N ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_331_Nat_Oex__has__greatest__nat,axiom,
% 3.05/1.58 ! [P3: nat > $o,K2: nat,B: nat] :
% 3.05/1.58 ( ( P3 @ K2 )
% 3.05/1.58 => ( ! [Y: nat] :
% 3.05/1.58 ( ( P3 @ Y )
% 3.05/1.58 => ( ord_less_eq_nat @ Y @ B ) )
% 3.05/1.58 => ? [X2: nat] :
% 3.05/1.58 ( ( P3 @ X2 )
% 3.05/1.58 & ! [Y5: nat] :
% 3.05/1.58 ( ( P3 @ Y5 )
% 3.05/1.58 => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_332_verit__la__generic,axiom,
% 3.05/1.58 ! [A: int,X: int] :
% 3.05/1.58 ( ( ord_less_eq_int @ A @ X )
% 3.05/1.58 | ( A = X )
% 3.05/1.58 | ( ord_less_eq_int @ X @ A ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_333_full__nat__induct,axiom,
% 3.05/1.58 ! [P3: nat > $o,N: nat] :
% 3.05/1.58 ( ! [N3: nat] :
% 3.05/1.58 ( ! [M3: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
% 3.05/1.58 => ( P3 @ M3 ) )
% 3.05/1.58 => ( P3 @ N3 ) )
% 3.05/1.58 => ( P3 @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_334_not__less__eq__eq,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 3.05/1.58 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_335_Suc__n__not__le__n,axiom,
% 3.05/1.58 ! [N: nat] :
% 3.05/1.58 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_336_nat__le__linear,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M @ N )
% 3.05/1.58 | ( ord_less_eq_nat @ N @ M ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_337_le__antisym,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M @ N )
% 3.05/1.58 => ( ( ord_less_eq_nat @ N @ M )
% 3.05/1.58 => ( M = N ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_338_le__Suc__eq,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 3.05/1.58 = ( ( ord_less_eq_nat @ M @ N )
% 3.05/1.58 | ( M
% 3.05/1.58 = ( suc @ N ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_339_eq__imp__le,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( M = N )
% 3.05/1.58 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_340_le__trans,axiom,
% 3.05/1.58 ! [I: nat,J: nat,K2: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ I @ J )
% 3.05/1.58 => ( ( ord_less_eq_nat @ J @ K2 )
% 3.05/1.58 => ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_341_Suc__le__D,axiom,
% 3.05/1.58 ! [N: nat,M4: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
% 3.05/1.58 => ? [M5: nat] :
% 3.05/1.58 ( M4
% 3.05/1.58 = ( suc @ M5 ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_342_le__refl,axiom,
% 3.05/1.58 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_343_le__SucI,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M @ N )
% 3.05/1.58 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(fact_344_le__SucE,axiom,
% 3.05/1.58 ! [M: nat,N: nat] :
% 3.05/1.58 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 3.05/1.58 => ( ~ ( ord_less_eq_nat @ M @ N )
% 3.05/1.58 => ( M
% 3.05/1.58 = ( suc @ N ) ) ) ) ).
% 3.05/1.58
% 3.05/1.58 thf(conj_0,conjecture,
% 3.05/1.58 ( ( sturm_469514713_r_pos @ p @ x )
% 3.05/1.58 != ( ~ ( ( ( ( poly_real2 @ p @ x )
% 3.05/1.58 = zero_zero_real )
% 3.05/1.58 => ( sturm_469514713_r_pos @ ( pderiv_real @ p ) @ x ) )
% 3.05/1.58 & ( ( ( poly_real2 @ p @ x )
% 3.05/1.58 != zero_zero_real )
% 3.05/1.58 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ p @ x ) ) ) ) ) ) ).
% 3.05/1.58 ------------------------
% 3.14/1.60 % (31904)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31714 for (2999ds/183Mi)
% 3.14/1.60 % (31907)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31714 for (2999ds/2Mi)
% 3.14/1.60 % (31905)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_31714 for (2999ds/4Mi)
% 3.14/1.60 % (31906)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_31714 for (2999ds/27Mi)
% 3.14/1.61 % (31908)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_31714 for (2999ds/2Mi)
% 3.14/1.61 % (31910)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31714 for (2999ds/18Mi)
% 3.14/1.61 % (31907)Instruction limit reached!
% 3.14/1.61 % (31907)------------------------------
% 3.14/1.61 % (31907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.61 % (31907)Termination reason: Unknown
% 3.14/1.61 % (31907)Termination phase: shuffling
% 3.14/1.61
% 3.14/1.61 % (31907)Memory used [KB]: 1407
% 3.14/1.61 % (31907)Time elapsed: 0.002 s
% 3.14/1.61 % (31907)Instructions burned: 2 (million)
% 3.14/1.61 % (31907)------------------------------
% 3.14/1.61 % (31907)------------------------------
% 3.14/1.61 % (31909)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_31714 for (2999ds/275Mi)
% 3.14/1.61 % (31908)Instruction limit reached!
% 3.14/1.61 % (31908)------------------------------
% 3.14/1.61 % (31908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.61 % (31908)Termination reason: Unknown
% 3.14/1.61 % (31908)Termination phase: shuffling
% 3.14/1.61
% 3.14/1.61 % (31908)Memory used [KB]: 1407
% 3.14/1.61 % (31908)Time elapsed: 0.003 s
% 3.14/1.61 % (31908)Instructions burned: 3 (million)
% 3.14/1.61 % (31908)------------------------------
% 3.14/1.61 % (31908)------------------------------
% 3.14/1.61 % (31905)Instruction limit reached!
% 3.14/1.61 % (31905)------------------------------
% 3.14/1.61 % (31905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.61 % (31905)Termination reason: Unknown
% 3.14/1.61 % (31905)Termination phase: shuffling
% 3.14/1.61
% 3.14/1.61 % (31905)Memory used [KB]: 1407
% 3.14/1.61 % (31905)Time elapsed: 0.004 s
% 3.14/1.61 % (31905)Instructions burned: 5 (million)
% 3.14/1.61 % (31905)------------------------------
% 3.14/1.61 % (31905)------------------------------
% 3.14/1.61 % (31910)Instruction limit reached!
% 3.14/1.61 % (31910)------------------------------
% 3.14/1.61 % (31910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.61 % (31910)Termination reason: Unknown
% 3.14/1.61 % (31910)Termination phase: shuffling
% 3.14/1.61
% 3.14/1.61 % (31910)Memory used [KB]: 1791
% 3.14/1.61 % (31910)Time elapsed: 0.011 s
% 3.14/1.61 % (31910)Instructions burned: 19 (million)
% 3.14/1.61 % (31910)------------------------------
% 3.14/1.61 % (31910)------------------------------
% 3.14/1.62 % (31906)Instruction limit reached!
% 3.14/1.62 % (31906)------------------------------
% 3.14/1.62 % (31906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.62 % (31906)Termination reason: Unknown
% 3.14/1.62 % (31906)Termination phase: Property scanning
% 3.14/1.62
% 3.14/1.62 % (31906)Memory used [KB]: 1918
% 3.14/1.62 % (31906)Time elapsed: 0.015 s
% 3.14/1.62 % (31906)Instructions burned: 28 (million)
% 3.14/1.62 % (31906)------------------------------
% 3.14/1.62 % (31906)------------------------------
% 3.14/1.62 % (31915)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_31714 for (2999ds/37Mi)
% 3.14/1.62 % (31913)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31714 for (2999ds/3Mi)
% 3.14/1.62 % (31913)Instruction limit reached!
% 3.14/1.62 % (31913)------------------------------
% 3.14/1.62 % (31913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.62 % (31913)Termination reason: Unknown
% 3.14/1.62 % (31913)Termination phase: shuffling
% 3.14/1.62
% 3.14/1.62 % (31913)Memory used [KB]: 1407
% 3.14/1.62 % (31913)Time elapsed: 0.004 s
% 3.14/1.62 % (31913)Instructions burned: 4 (million)
% 3.14/1.62 % (31913)------------------------------
% 3.14/1.62 % (31913)------------------------------
% 3.14/1.63 % (31916)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_31714 for (2999ds/15Mi)
% 3.14/1.63 % (31917)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_31714 for (2999ds/3Mi)
% 3.14/1.63 % (31917)Instruction limit reached!
% 3.14/1.63 % (31917)------------------------------
% 3.14/1.63 % (31917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.63 % (31917)Termination reason: Unknown
% 3.14/1.63 % (31917)Termination phase: shuffling
% 3.14/1.63
% 3.14/1.63 % (31917)Memory used [KB]: 1407
% 3.14/1.63 % (31917)Time elapsed: 0.003 s
% 3.14/1.63 % (31917)Instructions burned: 3 (million)
% 3.14/1.63 % (31917)------------------------------
% 3.14/1.63 % (31917)------------------------------
% 3.14/1.64 % (31916)Instruction limit reached!
% 3.14/1.64 % (31916)------------------------------
% 3.14/1.64 % (31916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.64 % (31916)Termination reason: Unknown
% 3.14/1.64 % (31916)Termination phase: shuffling
% 3.14/1.64
% 3.14/1.64 % (31916)Memory used [KB]: 1663
% 3.14/1.64 % (31916)Time elapsed: 0.009 s
% 3.14/1.64 % (31916)Instructions burned: 15 (million)
% 3.14/1.64 % (31916)------------------------------
% 3.14/1.64 % (31916)------------------------------
% 3.14/1.64 % (31915)Instruction limit reached!
% 3.14/1.64 % (31915)------------------------------
% 3.14/1.64 % (31915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.64 % (31915)Termination reason: Unknown
% 3.14/1.64 % (31915)Termination phase: shuffling
% 3.14/1.64
% 3.14/1.64 % (31915)Memory used [KB]: 2174
% 3.14/1.64 % (31915)Time elapsed: 0.018 s
% 3.14/1.64 % (31915)Instructions burned: 37 (million)
% 3.14/1.64 % (31915)------------------------------
% 3.14/1.64 % (31915)------------------------------
% 3.14/1.64 % (31909)First to succeed.
% 3.14/1.64 % (31918)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_31714 for (2999ds/1041Mi)
% 3.14/1.64 % (31922)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31714 for (2999ds/7Mi)
% 3.14/1.64 % (31909)Refutation found. Thanks to Tanya!
% 3.14/1.64 % SZS status Theorem for DTF2THF_31714
% 3.14/1.64 % SZS output start Proof for DTF2THF_31714
% 3.14/1.64 thf(type_def_5, type, nat: $tType).
% 3.14/1.64 thf(type_def_6, type, int: $tType).
% 3.14/1.64 thf(type_def_7, type, poly_int: $tType).
% 3.14/1.64 thf(type_def_8, type, poly_nat: $tType).
% 3.14/1.64 thf(type_def_9, type, poly_poly_int: $tType).
% 3.14/1.64 thf(type_def_10, type, poly_poly_nat: $tType).
% 3.14/1.64 thf(type_def_11, type, poly_poly_poly_real: $tType).
% 3.14/1.64 thf(type_def_12, type, poly_poly_real: $tType).
% 3.14/1.64 thf(type_def_13, type, poly_real: $tType).
% 3.14/1.64 thf(type_def_14, type, real: $tType).
% 3.14/1.64 thf(type_def_16, type, set_real: $tType).
% 3.14/1.64 thf(func_def_0, type, poly_poly_poly_real: $tType).
% 3.14/1.64 thf(func_def_1, type, poly_poly_real: $tType).
% 3.14/1.64 thf(func_def_2, type, poly_poly_nat: $tType).
% 3.14/1.64 thf(func_def_3, type, poly_poly_int: $tType).
% 3.14/1.64 thf(func_def_4, type, poly_real: $tType).
% 3.14/1.64 thf(func_def_5, type, poly_nat: $tType).
% 3.14/1.64 thf(func_def_6, type, poly_int: $tType).
% 3.14/1.64 thf(func_def_7, type, set_real: $tType).
% 3.14/1.64 thf(func_def_8, type, real: $tType).
% 3.14/1.64 thf(func_def_9, type, nat: $tType).
% 3.14/1.64 thf(func_def_10, type, int: $tType).
% 3.14/1.64 thf(func_def_11, type, one_one_nat: nat).
% 3.14/1.64 thf(func_def_12, type, zero_zero_int: int).
% 3.14/1.64 thf(func_def_13, type, zero_zero_nat: nat).
% 3.14/1.64 thf(func_def_14, type, zero_zero_poly_int: poly_int).
% 3.14/1.64 thf(func_def_15, type, zero_zero_poly_nat: poly_nat).
% 3.14/1.64 thf(func_def_16, type, zero_z1549157189ly_int: poly_poly_int).
% 3.14/1.64 thf(func_def_17, type, zero_z1059985641ly_nat: poly_poly_nat).
% 3.14/1.64 thf(func_def_18, type, zero_z935034829y_real: poly_poly_poly_real).
% 3.14/1.64 thf(func_def_19, type, zero_z1423781445y_real: poly_poly_real).
% 3.14/1.64 thf(func_def_20, type, zero_zero_poly_real: poly_real).
% 3.14/1.64 thf(func_def_21, type, zero_zero_real: real).
% 3.14/1.64 thf(func_def_22, type, suc: nat > nat).
% 3.14/1.64 thf(func_def_23, type, ord_less_int: int > int > $o).
% 3.14/1.64 thf(func_def_24, type, ord_less_nat: nat > nat > $o).
% 3.14/1.64 thf(func_def_25, type, ord_less_poly_int: poly_int > poly_int > $o).
% 3.14/1.64 thf(func_def_26, type, ord_le38482960y_real: poly_poly_real > poly_poly_real > $o).
% 3.14/1.64 thf(func_def_27, type, ord_less_poly_real: poly_real > poly_real > $o).
% 3.14/1.64 thf(func_def_28, type, ord_less_real: real > real > $o).
% 3.14/1.64 thf(func_def_29, type, ord_less_eq_int: int > int > $o).
% 3.14/1.64 thf(func_def_30, type, ord_less_eq_nat: nat > nat > $o).
% 3.14/1.64 thf(func_def_31, type, ord_less_eq_poly_int: poly_int > poly_int > $o).
% 3.14/1.64 thf(func_def_32, type, ord_le893774876y_real: poly_poly_real > poly_poly_real > $o).
% 3.14/1.64 thf(func_def_33, type, ord_le1180086932y_real: poly_real > poly_real > $o).
% 3.14/1.64 thf(func_def_34, type, ord_less_eq_real: real > real > $o).
% 3.14/1.64 thf(func_def_35, type, is_zero_int: poly_int > $o).
% 3.14/1.64 thf(func_def_36, type, is_zero_nat: poly_nat > $o).
% 3.14/1.64 thf(func_def_37, type, is_zero_poly_real: poly_poly_real > $o).
% 3.14/1.64 thf(func_def_38, type, is_zero_real: poly_real > $o).
% 3.14/1.64 thf(func_def_39, type, order_int: int > poly_int > nat).
% 3.14/1.64 thf(func_def_40, type, order_poly_int: poly_int > poly_poly_int > nat).
% 3.14/1.64 thf(func_def_41, type, order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat).
% 3.14/1.64 thf(func_def_42, type, order_poly_real: poly_real > poly_poly_real > nat).
% 3.14/1.64 thf(func_def_43, type, order_real: real > poly_real > nat).
% 3.14/1.64 thf(func_def_44, type, pderiv_int: poly_int > poly_int).
% 3.14/1.64 thf(func_def_45, type, pderiv_nat: poly_nat > poly_nat).
% 3.14/1.64 thf(func_def_46, type, pderiv_poly_real: poly_poly_real > poly_poly_real).
% 3.14/1.64 thf(func_def_47, type, pderiv_real: poly_real > poly_real).
% 3.14/1.64 thf(func_def_48, type, poly_int2: poly_int > int > int).
% 3.14/1.64 thf(func_def_49, type, poly_nat2: poly_nat > nat > nat).
% 3.14/1.64 thf(func_def_50, type, poly_poly_int2: poly_poly_int > poly_int > poly_int).
% 3.14/1.64 thf(func_def_51, type, poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat).
% 3.14/1.64 thf(func_def_52, type, poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real).
% 3.14/1.64 thf(func_def_53, type, poly_poly_real2: poly_poly_real > poly_real > poly_real).
% 3.14/1.64 thf(func_def_54, type, poly_real2: poly_real > real > real).
% 3.14/1.64 thf(func_def_55, type, poly_cutoff_int: nat > poly_int > poly_int).
% 3.14/1.64 thf(func_def_56, type, poly_cutoff_nat: nat > poly_nat > poly_nat).
% 3.14/1.64 thf(func_def_57, type, poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real).
% 3.14/1.64 thf(func_def_58, type, poly_cutoff_real: nat > poly_real > poly_real).
% 3.14/1.64 thf(func_def_59, type, poly_shift_int: nat > poly_int > poly_int).
% 3.14/1.64 thf(func_def_60, type, poly_shift_nat: nat > poly_nat > poly_nat).
% 3.14/1.64 thf(func_def_61, type, poly_shift_poly_real: nat > poly_poly_real > poly_poly_real).
% 3.14/1.64 thf(func_def_62, type, poly_shift_real: nat > poly_real > poly_real).
% 3.14/1.64 thf(func_def_63, type, reflect_poly_int: poly_int > poly_int).
% 3.14/1.64 thf(func_def_64, type, reflect_poly_nat: poly_nat > poly_nat).
% 3.14/1.64 thf(func_def_65, type, reflec943087998ly_int: poly_poly_int > poly_poly_int).
% 3.14/1.64 thf(func_def_66, type, reflec781175074ly_nat: poly_poly_nat > poly_poly_nat).
% 3.14/1.64 thf(func_def_67, type, reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real).
% 3.14/1.64 thf(func_def_68, type, reflec1522834046y_real: poly_poly_real > poly_poly_real).
% 3.14/1.64 thf(func_def_69, type, reflect_poly_real: poly_real > poly_real).
% 3.14/1.64 thf(func_def_70, type, rsquarefree_real: poly_real > $o).
% 3.14/1.64 thf(func_def_71, type, collect_real: (real > $o) > set_real).
% 3.14/1.64 thf(func_def_72, type, sturm_1953858694_cross: poly_real > real > real > int).
% 3.14/1.64 thf(func_def_73, type, sturm_469514713_r_pos: poly_real > real > $o).
% 3.14/1.64 thf(func_def_74, type, sturm_1279781401iation: real > real > int).
% 3.14/1.64 thf(func_def_75, type, member_real: real > set_real > $o).
% 3.14/1.64 thf(func_def_76, type, p: poly_real).
% 3.14/1.64 thf(func_def_77, type, x: real).
% 3.14/1.64 thf(func_def_92, type, sP0: real > real > real > $o).
% 3.14/1.64 thf(func_def_93, type, sP1: real > real > real > $o).
% 3.14/1.64 thf(func_def_94, type, sP2: real > real > real > $o).
% 3.14/1.64 thf(func_def_95, type, sK3: (nat > nat) > nat).
% 3.14/1.64 thf(func_def_96, type, sK4: (nat > nat) > nat).
% 3.14/1.64 thf(func_def_97, type, sK5: real > real).
% 3.14/1.64 thf(func_def_98, type, sK6: (nat > real) > nat).
% 3.14/1.64 thf(func_def_99, type, sK7: (nat > real) > nat).
% 3.14/1.64 thf(func_def_100, type, sK8: (nat > real) > nat).
% 3.14/1.64 thf(func_def_101, type, sK9: (nat > real) > nat).
% 3.14/1.64 thf(func_def_102, type, sK10: (real > real) > real).
% 3.14/1.64 thf(func_def_103, type, sK11: (real > real) > real).
% 3.14/1.64 thf(func_def_104, type, sK12: (nat > real) > nat).
% 3.14/1.64 thf(func_def_105, type, sK13: (nat > real) > nat).
% 3.14/1.64 thf(func_def_106, type, sK14: real > real).
% 3.14/1.64 thf(func_def_107, type, sK15: real > poly_real > real).
% 3.14/1.64 thf(func_def_108, type, sK16: (real > real) > real).
% 3.14/1.64 thf(func_def_109, type, sK17: (real > real) > real).
% 3.14/1.64 thf(func_def_110, type, sK18: (real > real > $o) > real).
% 3.14/1.64 thf(func_def_111, type, sK19: (real > real > $o) > real).
% 3.14/1.64 thf(func_def_112, type, sK20: (real > real > $o) > real).
% 3.14/1.64 thf(func_def_113, type, sK21: (real > real > $o) > real).
% 3.14/1.64 thf(func_def_114, type, sK22: (real > real > $o) > real).
% 3.14/1.64 thf(func_def_115, type, sK23: (nat > nat) > nat).
% 3.14/1.64 thf(func_def_116, type, sK24: (nat > nat) > nat).
% 3.14/1.64 thf(func_def_117, type, sK25: (real > nat) > real).
% 3.14/1.64 thf(func_def_118, type, sK26: (real > nat) > real).
% 3.14/1.64 thf(func_def_119, type, sK27: real > real > real).
% 3.14/1.64 thf(func_def_120, type, sK28: poly_real > real > real > real).
% 3.14/1.64 thf(func_def_121, type, sK29: (real > real) > real).
% 3.14/1.64 thf(func_def_122, type, sK30: (real > real) > real).
% 3.14/1.64 thf(func_def_123, type, sK31: (nat > real) > nat).
% 3.14/1.64 thf(func_def_124, type, sK32: (nat > real) > nat).
% 3.14/1.64 thf(func_def_125, type, sK33: (real > nat) > real).
% 3.14/1.64 thf(func_def_126, type, sK34: (real > nat) > real).
% 3.14/1.64 thf(func_def_127, type, sK35: real > real).
% 3.14/1.64 thf(func_def_128, type, sK36: (real > nat) > real).
% 3.14/1.64 thf(func_def_129, type, sK37: (real > nat) > real).
% 3.14/1.64 thf(func_def_130, type, sK38: (nat > real) > nat).
% 3.14/1.64 thf(func_def_131, type, sK39: (nat > real) > nat).
% 3.14/1.64 thf(func_def_132, type, sK40: nat > nat).
% 3.14/1.64 thf(func_def_133, type, sK41: (nat > real) > nat).
% 3.14/1.64 thf(func_def_134, type, sK42: (nat > real) > nat).
% 3.14/1.64 thf(func_def_135, type, sK43: (real > real > $o) > real).
% 3.14/1.64 thf(func_def_136, type, sK44: (real > real > $o) > real).
% 3.14/1.64 thf(func_def_137, type, sK45: (real > real > $o) > real).
% 3.14/1.64 thf(func_def_138, type, sK46: (real > real > $o) > real).
% 3.14/1.64 thf(func_def_139, type, sK47: (real > nat) > real).
% 3.14/1.64 thf(func_def_140, type, sK48: (real > nat) > real).
% 3.14/1.64 thf(func_def_141, type, sK49: (real > nat) > real).
% 3.14/1.64 thf(func_def_142, type, sK50: (real > nat) > real).
% 3.14/1.64 thf(func_def_143, type, sK51: (real > nat) > real).
% 3.14/1.64 thf(func_def_144, type, sK52: (real > nat) > real).
% 3.14/1.64 thf(func_def_145, type, sK53: (nat > $o) > nat).
% 3.14/1.64 thf(func_def_146, type, sK54: real > real).
% 3.14/1.64 thf(func_def_147, type, sK55: real > real).
% 3.14/1.64 thf(func_def_148, type, sK56: poly_real > real).
% 3.14/1.64 thf(func_def_149, type, sK57: (nat > $o) > nat).
% 3.14/1.64 thf(func_def_150, type, sK58: poly_real > real > real > real).
% 3.14/1.64 thf(func_def_151, type, sK59: (real > nat) > real).
% 3.14/1.64 thf(func_def_152, type, sK60: (real > nat) > real).
% 3.14/1.64 thf(func_def_153, type, sK61: poly_real > real > real > real).
% 3.14/1.64 thf(func_def_154, type, sK62: poly_real > real > real > real).
% 3.14/1.64 thf(func_def_155, type, sK63: poly_real > real > real > real).
% 3.14/1.64 thf(func_def_156, type, sK64: (nat > nat) > nat).
% 3.14/1.64 thf(func_def_157, type, sK65: (nat > nat) > nat).
% 3.14/1.64 thf(func_def_158, type, sK66: poly_real > real > real).
% 3.14/1.64 thf(func_def_159, type, sK67: real > real).
% 3.14/1.64 thf(func_def_160, type, sK68: (real > nat) > real).
% 3.14/1.64 thf(func_def_161, type, sK69: (real > nat) > real).
% 3.14/1.64 thf(func_def_162, type, sK70: (real > real) > real).
% 3.14/1.64 thf(func_def_163, type, sK71: (real > real) > real).
% 3.14/1.64 thf(func_def_164, type, sK72: (nat > real) > nat).
% 3.14/1.64 thf(func_def_165, type, sK73: (nat > real) > nat).
% 3.14/1.64 thf(func_def_166, type, sK74: real > real > real).
% 3.14/1.64 thf(func_def_167, type, sK75: (real > real) > real).
% 3.14/1.64 thf(func_def_168, type, sK76: (real > real) > real).
% 3.14/1.64 thf(func_def_169, type, sK77: (nat > nat > $o) > nat).
% 3.14/1.64 thf(func_def_170, type, sK78: (nat > nat > $o) > nat).
% 3.14/1.64 thf(func_def_171, type, sK79: (nat > nat > $o) > nat).
% 3.14/1.64 thf(func_def_172, type, sK80: (nat > nat > $o) > nat).
% 3.14/1.64 thf(func_def_173, type, sK81: (nat > nat > $o) > nat).
% 3.14/1.64 thf(func_def_174, type, sK82: (real > real) > real).
% 3.14/1.64 thf(func_def_175, type, sK83: (real > real) > real).
% 3.14/1.64 thf(func_def_176, type, sK84: (nat > nat) > nat).
% 3.14/1.64 thf(func_def_177, type, sK85: (nat > nat) > nat).
% 3.14/1.64 thf(func_def_178, type, sK86: (nat > real) > nat).
% 3.14/1.64 thf(func_def_179, type, sK87: (nat > real) > nat).
% 3.14/1.64 thf(func_def_180, type, sK88: (real > real) > real).
% 3.14/1.64 thf(func_def_181, type, sK89: (real > real) > real).
% 3.14/1.64 thf(func_def_182, type, sK90: (nat > $o) > nat).
% 3.14/1.64 thf(func_def_183, type, sK91: poly_real > real > real > real).
% 3.14/1.64 thf(func_def_184, type, sK92: poly_real > real > real > real).
% 3.14/1.64 thf(func_def_185, type, sK93: poly_real > real > real > real).
% 3.14/1.64 thf(func_def_186, type, sK94: (real > real) > real).
% 3.14/1.64 thf(func_def_187, type, sK95: (real > real) > real).
% 3.14/1.64 thf(func_def_189, type, ph97: !>[X0: $tType]:(X0)).
% 3.14/1.64 thf(f1651,plain,(
% 3.14/1.64 $false),
% 3.14/1.64 inference(subsumption_resolution,[status(thm)],[f1632,f1618])).
% 3.14/1.64 thf(f1618,plain,(
% 3.14/1.64 ((sturm_469514713_r_pos @ p @ x) != $true)),
% 3.14/1.64 inference(subsumption_resolution,[status(thm)],[f1617,f1616])).
% 3.14/1.64 thf(f1616,plain,(
% 3.14/1.64 ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true)),
% 3.14/1.64 inference(subsumption_resolution,[status(thm)],[f1615,f1530])).
% 3.14/1.64 thf(f1530,plain,(
% 3.14/1.64 ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true) | ((sturm_469514713_r_pos @ p @ x) != $true)),
% 3.14/1.64 inference(cnf_transformation,[status(thm)],[f1072])).
% 3.14/1.64 thf(f1072,plain,(
% 3.14/1.64 ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true) | ((sturm_469514713_r_pos @ p @ x) != $true)),
% 3.14/1.64 inference(ennf_transformation,[status(thm)],[f953])).
% 3.14/1.64 thf(f953,plain,(
% 3.14/1.64 ((sturm_469514713_r_pos @ p @ x) = $true) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true)),
% 3.14/1.64 inference(fool_elimination,[status(thm)],[f952])).
% 3.14/1.64 thf(f952,plain,(
% 3.14/1.64 (sturm_469514713_r_pos @ p @ x) => (ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x))),
% 3.14/1.64 inference(rectify,[status(thm)],[f2])).
% 3.14/1.64 thf(f2,axiom,(
% 3.14/1.64 (sturm_469514713_r_pos @ p @ x) => (ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x))),
% 3.14/1.64 file('/export/starexec/sandbox2/tmp/tmp.JHQKkcEisk/DTF2THF_31714.p',fact_1__092_060open_062sign__r__pos_Ap_Ax_A_092_060Longrightarrow_062_A0_A_060_Apoly_Ap_Ax_092_060close_062)).
% 3.14/1.64 thf(f1615,plain,(
% 3.14/1.64 ((sturm_469514713_r_pos @ p @ x) = $true) | ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true)),
% 3.14/1.64 inference(subsumption_resolution,[status(thm)],[f1353,f1510])).
% 3.14/1.64 thf(f1510,plain,(
% 3.14/1.64 ((poly_real2 @ p @ x) != zero_zero_real)),
% 3.14/1.64 inference(cnf_transformation,[status(thm)],[f1])).
% 3.14/1.64 thf(f1,axiom,(
% 3.14/1.64 ((poly_real2 @ p @ x) != zero_zero_real)),
% 3.14/1.64 file('/export/starexec/sandbox2/tmp/tmp.JHQKkcEisk/DTF2THF_31714.p',fact_0_False)).
% 3.14/1.64 thf(f1353,plain,(
% 3.14/1.64 ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true) | ((sturm_469514713_r_pos @ p @ x) = $true) | ((poly_real2 @ p @ x) = zero_zero_real)),
% 3.14/1.64 inference(cnf_transformation,[status(thm)],[f1191])).
% 3.14/1.64 thf(f1191,plain,(
% 3.14/1.64 ((((sturm_469514713_r_pos @ (pderiv_real @ p) @ x) != $true) & ((poly_real2 @ p @ x) = zero_zero_real)) | (((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) != $true) & ((poly_real2 @ p @ x) != zero_zero_real)) | ((sturm_469514713_r_pos @ p @ x) != $true)) & (((sturm_469514713_r_pos @ p @ x) = $true) | ((((sturm_469514713_r_pos @ (pderiv_real @ p) @ x) = $true) | ((poly_real2 @ p @ x) != zero_zero_real)) & (((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true) | ((poly_real2 @ p @ x) = zero_zero_real))))),
% 3.14/1.64 inference(flattening,[status(thm)],[f1190])).
% 3.14/1.64 thf(f1190,plain,(
% 3.14/1.64 (((((sturm_469514713_r_pos @ (pderiv_real @ p) @ x) != $true) & ((poly_real2 @ p @ x) = zero_zero_real)) | (((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) != $true) & ((poly_real2 @ p @ x) != zero_zero_real))) | ((sturm_469514713_r_pos @ p @ x) != $true)) & (((sturm_469514713_r_pos @ p @ x) = $true) | ((((sturm_469514713_r_pos @ (pderiv_real @ p) @ x) = $true) | ((poly_real2 @ p @ x) != zero_zero_real)) & (((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true) | ((poly_real2 @ p @ x) = zero_zero_real))))),
% 3.14/1.64 inference(nnf_transformation,[status(thm)],[f1157])).
% 3.14/1.64 thf(f1157,plain,(
% 3.14/1.64 ((((sturm_469514713_r_pos @ (pderiv_real @ p) @ x) != $true) & ((poly_real2 @ p @ x) = zero_zero_real)) | (((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) != $true) & ((poly_real2 @ p @ x) != zero_zero_real))) <=> ((sturm_469514713_r_pos @ p @ x) = $true)),
% 3.14/1.64 inference(ennf_transformation,[status(thm)],[f625])).
% 3.14/1.64 thf(f625,plain,(
% 3.14/1.64 ~(~((((poly_real2 @ p @ x) = zero_zero_real) => ((sturm_469514713_r_pos @ (pderiv_real @ p) @ x) = $true)) & (((poly_real2 @ p @ x) != zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true))) <~> ((sturm_469514713_r_pos @ p @ x) = $true))),
% 3.14/1.64 inference(fool_elimination,[status(thm)],[f624])).
% 3.14/1.64 thf(f624,plain,(
% 3.14/1.64 ~((sturm_469514713_r_pos @ p @ x) != ~((((poly_real2 @ p @ x) != zero_zero_real) => (ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x))) & (((poly_real2 @ p @ x) = zero_zero_real) => (sturm_469514713_r_pos @ (pderiv_real @ p) @ x))))),
% 3.14/1.64 inference(rectify,[status(thm)],[f347])).
% 3.14/1.64 thf(f347,negated_conjecture,(
% 3.14/1.64 ~((sturm_469514713_r_pos @ p @ x) != ~((((poly_real2 @ p @ x) != zero_zero_real) => (ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x))) & (((poly_real2 @ p @ x) = zero_zero_real) => (sturm_469514713_r_pos @ (pderiv_real @ p) @ x))))),
% 3.14/1.64 inference(negated_conjecture,[status(cth)],[f346])).
% 3.14/1.64 thf(f346,conjecture,(
% 3.14/1.64 ((sturm_469514713_r_pos @ p @ x) != ~((((poly_real2 @ p @ x) != zero_zero_real) => (ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x))) & (((poly_real2 @ p @ x) = zero_zero_real) => (sturm_469514713_r_pos @ (pderiv_real @ p) @ x))))),
% 3.14/1.64 file('/export/starexec/sandbox2/tmp/tmp.JHQKkcEisk/DTF2THF_31714.p',conj_0)).
% 3.14/1.64 thf(f1617,plain,(
% 3.14/1.64 ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) != $true) | ((sturm_469514713_r_pos @ p @ x) != $true)),
% 3.14/1.64 inference(subsumption_resolution,[status(thm)],[f1356,f1510])).
% 3.14/1.64 thf(f1356,plain,(
% 3.14/1.64 ((sturm_469514713_r_pos @ p @ x) != $true) | ((poly_real2 @ p @ x) = zero_zero_real) | ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) != $true)),
% 3.14/1.64 inference(cnf_transformation,[status(thm)],[f1191])).
% 3.14/1.64 thf(f1632,plain,(
% 3.14/1.64 ((sturm_469514713_r_pos @ p @ x) = $true)),
% 3.14/1.64 inference(trivial_inequality_removal,[status(thm)],[f1619])).
% 3.14/1.64 thf(f1619,plain,(
% 3.14/1.64 ((sturm_469514713_r_pos @ p @ x) = $true) | ($true != $true)),
% 3.14/1.64 inference(superposition,[status(thm)],[f1342,f1616])).
% 3.14/1.64 thf(f1342,plain,(
% 3.14/1.64 ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) != $true) | ((sturm_469514713_r_pos @ p @ x) = $true)),
% 3.14/1.64 inference(cnf_transformation,[status(thm)],[f1037])).
% 3.14/1.64 thf(f1037,plain,(
% 3.14/1.64 ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) != $true) | ((sturm_469514713_r_pos @ p @ x) = $true)),
% 3.14/1.64 inference(ennf_transformation,[status(thm)],[f464])).
% 3.14/1.64 thf(f464,plain,(
% 3.14/1.64 ((ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) = $true) => ((sturm_469514713_r_pos @ p @ x) = $true)),
% 3.14/1.64 inference(fool_elimination,[status(thm)],[f463])).
% 3.14/1.64 thf(f463,plain,(
% 3.14/1.64 (ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) => (sturm_469514713_r_pos @ p @ x)),
% 3.14/1.64 inference(rectify,[status(thm)],[f3])).
% 3.14/1.64 thf(f3,axiom,(
% 3.14/1.64 (ord_less_real @ zero_zero_real @ (poly_real2 @ p @ x)) => (sturm_469514713_r_pos @ p @ x)),
% 3.14/1.64 file('/export/starexec/sandbox2/tmp/tmp.JHQKkcEisk/DTF2THF_31714.p',fact_2__092_060open_0620_A_060_Apoly_Ap_Ax_A_092_060Longrightarrow_062_Asign__r__pos_Ap_Ax_092_060close_062)).
% 3.14/1.64 % SZS output end Proof for DTF2THF_31714
% 3.14/1.64 % (31909)------------------------------
% 3.14/1.64 % (31909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.64 % (31909)Termination reason: Refutation
% 3.14/1.64
% 3.14/1.64 % (31909)Memory used [KB]: 6908
% 3.14/1.64 % (31909)Time elapsed: 0.037 s
% 3.14/1.64 % (31909)Instructions burned: 88 (million)
% 3.14/1.64 % (31909)------------------------------
% 3.14/1.64 % (31909)------------------------------
% 3.14/1.64 % (31897)Success in time 0.057 s
%------------------------------------------------------------------------------