%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP194^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n018.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 3.13s 1.74s
% Output : Refutation 3.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 5
% Syntax : Number of formulae : 27 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 90 ( 38 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 148 ( 21 ~; 10 |; 4 &; 106 @)
% ( 0 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 196 ( 193 usr; 15 con; 0-6 aty)
% Number of variables : 27 ( 0 ^; 25 !; 2 ?; 27 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
poly_poly_real: $tType ).
thf(type_def_7,type,
poly_real: $tType ).
thf(type_def_8,type,
real: $tType ).
thf(type_def_9,type,
nat: $tType ).
thf(type_def_10,type,
poly_nat: $tType ).
thf(type_def_11,type,
poly_poly_nat: $tType ).
thf(type_def_12,type,
poly_poly_poly_real: $tType ).
thf(type_def_13,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_real: $tType ).
thf(func_def_4,type,
poly_nat: $tType ).
thf(func_def_5,type,
set_real: $tType ).
thf(func_def_6,type,
real: $tType ).
thf(func_def_7,type,
nat: $tType ).
thf(func_def_8,type,
sgn_sg2128174761y_real: poly_poly_real > poly_poly_real ).
thf(func_def_9,type,
sgn_sgn_poly_real: poly_real > poly_real ).
thf(func_def_10,type,
sgn_sgn_real: real > real ).
thf(func_def_11,type,
zero_zero_nat: nat ).
thf(func_def_12,type,
zero_zero_poly_nat: poly_nat ).
thf(func_def_13,type,
zero_z1059985641ly_nat: poly_poly_nat ).
thf(func_def_14,type,
zero_z935034829y_real: poly_poly_poly_real ).
thf(func_def_15,type,
zero_z1423781445y_real: poly_poly_real ).
thf(func_def_16,type,
zero_zero_poly_real: poly_real ).
thf(func_def_17,type,
zero_zero_real: real ).
thf(func_def_18,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_19,type,
if_real: $o > real > real > real ).
thf(func_def_20,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_21,type,
ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
thf(func_def_22,type,
ord_less_poly_real: poly_real > poly_real > $o ).
thf(func_def_23,type,
ord_less_real: real > real > $o ).
thf(func_def_24,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_25,type,
ord_le893774876y_real: poly_poly_real > poly_poly_real > $o ).
thf(func_def_26,type,
ord_le1180086932y_real: poly_real > poly_real > $o ).
thf(func_def_27,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_28,type,
ord_min_nat: nat > nat > nat ).
thf(func_def_29,type,
ord_min_real: real > real > real ).
thf(func_def_30,type,
divide924636027y_real: poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > nat > nat > poly_poly_poly_real ).
thf(func_def_31,type,
divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real ).
thf(func_def_32,type,
divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real ).
thf(func_def_33,type,
is_zero_nat: poly_nat > $o ).
thf(func_def_34,type,
is_zero_poly_real: poly_poly_real > $o ).
thf(func_def_35,type,
is_zero_real: poly_real > $o ).
thf(func_def_36,type,
order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
thf(func_def_37,type,
order_poly_real: poly_real > poly_poly_real > nat ).
thf(func_def_38,type,
order_real: real > poly_real > nat ).
thf(func_def_39,type,
poly_nat2: poly_nat > nat > nat ).
thf(func_def_40,type,
poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
thf(func_def_41,type,
poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
thf(func_def_42,type,
poly_poly_real2: poly_poly_real > poly_real > poly_real ).
thf(func_def_43,type,
poly_real2: poly_real > real > real ).
thf(func_def_44,type,
poly_cutoff_nat: nat > poly_nat > poly_nat ).
thf(func_def_45,type,
poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
thf(func_def_46,type,
poly_cutoff_real: nat > poly_real > poly_real ).
thf(func_def_47,type,
poly_shift_nat: nat > poly_nat > poly_nat ).
thf(func_def_48,type,
poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
thf(func_def_49,type,
poly_shift_real: nat > poly_real > poly_real ).
thf(func_def_50,type,
reflect_poly_nat: poly_nat > poly_nat ).
thf(func_def_51,type,
reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
thf(func_def_52,type,
reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
thf(func_def_53,type,
reflec1522834046y_real: poly_poly_real > poly_poly_real ).
thf(func_def_54,type,
reflect_poly_real: poly_real > poly_real ).
thf(func_def_55,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_56,type,
sturm_1076696862f_real: poly_real > real ).
thf(func_def_57,type,
sturm_1308388506f_real: poly_real > real ).
thf(func_def_58,type,
member_real: real > set_real > $o ).
thf(func_def_59,type,
lb1: real ).
thf(func_def_60,type,
lb2: real ).
thf(func_def_61,type,
lb: real ).
thf(func_def_62,type,
p: poly_real ).
thf(func_def_78,type,
sP0: real > real > real > $o ).
thf(func_def_79,type,
sP1: real > real > real > $o ).
thf(func_def_80,type,
sP2: real > real > real > $o ).
thf(func_def_81,type,
sP3: nat > nat > nat > $o ).
thf(func_def_82,type,
sP4: nat > nat > nat > $o ).
thf(func_def_83,type,
sP5: nat > nat > nat > $o ).
thf(func_def_84,type,
sK6: ( real > nat ) > real ).
thf(func_def_85,type,
sK7: ( real > nat ) > real ).
thf(func_def_86,type,
sK8: poly_real > real > real > real ).
thf(func_def_87,type,
sK9: nat > ( nat > $o ) > nat ).
thf(func_def_88,type,
sK10: ( nat > $o ) > nat ).
thf(func_def_89,type,
sK11: ( real > real ) > real ).
thf(func_def_90,type,
sK12: ( real > real ) > real ).
thf(func_def_91,type,
sK13: ( real > real ) > real ).
thf(func_def_92,type,
sK14: ( real > real ) > real ).
thf(func_def_93,type,
sK15: ( nat > nat ) > nat ).
thf(func_def_94,type,
sK16: ( nat > nat ) > nat ).
thf(func_def_95,type,
sK17: ( nat > nat ) > nat ).
thf(func_def_96,type,
sK18: ( nat > nat ) > nat ).
thf(func_def_97,type,
sK19: real > poly_real > real ).
thf(func_def_98,type,
sK20: ( nat > nat ) > nat ).
thf(func_def_99,type,
sK21: ( nat > nat ) > nat ).
thf(func_def_100,type,
sK22: ( real > real ) > real ).
thf(func_def_101,type,
sK23: ( real > real ) > real ).
thf(func_def_102,type,
sK24: real > real > real ).
thf(func_def_103,type,
sK25: ( real > real ) > real ).
thf(func_def_104,type,
sK26: ( real > real ) > real ).
thf(func_def_105,type,
sK27: ( nat > real ) > nat ).
thf(func_def_106,type,
sK28: ( nat > real ) > nat ).
thf(func_def_107,type,
sK29: ( nat > nat > $o ) > nat ).
thf(func_def_108,type,
sK30: ( nat > nat > $o ) > nat ).
thf(func_def_109,type,
sK31: ( nat > nat > $o ) > nat ).
thf(func_def_110,type,
sK32: ( nat > nat > $o ) > nat ).
thf(func_def_111,type,
sK33: ( nat > nat > $o ) > nat ).
thf(func_def_112,type,
sK34: ( nat > real ) > nat ).
thf(func_def_113,type,
sK35: ( nat > real ) > nat ).
thf(func_def_114,type,
sK36: ( nat > $o ) > nat ).
thf(func_def_115,type,
sK37: poly_real > real ).
thf(func_def_116,type,
sK38: ( real > real ) > real ).
thf(func_def_117,type,
sK39: ( real > real ) > real ).
thf(func_def_118,type,
sK40: ( nat > real ) > nat ).
thf(func_def_119,type,
sK41: ( nat > real ) > nat ).
thf(func_def_120,type,
sK42: ( real > nat ) > real ).
thf(func_def_121,type,
sK43: ( real > nat ) > real ).
thf(func_def_122,type,
sK44: real ).
thf(func_def_123,type,
sK45: ( nat > nat ) > nat ).
thf(func_def_124,type,
sK46: ( nat > nat ) > nat ).
thf(func_def_125,type,
sK47: ( nat > real ) > nat ).
thf(func_def_126,type,
sK48: ( nat > real ) > nat ).
thf(func_def_127,type,
sK49: ( real > real ) > real ).
thf(func_def_128,type,
sK50: ( real > real ) > real ).
thf(func_def_129,type,
sK51: ( nat > nat ) > nat ).
thf(func_def_130,type,
sK52: ( nat > nat ) > nat ).
thf(func_def_131,type,
sK53: ( nat > nat ) > nat ).
thf(func_def_132,type,
sK54: ( nat > nat ) > nat ).
thf(func_def_133,type,
sK55: ( real > nat ) > real ).
thf(func_def_134,type,
sK56: ( real > nat ) > real ).
thf(func_def_135,type,
sK57: ( real > real ) > real ).
thf(func_def_136,type,
sK58: ( real > real ) > real ).
thf(func_def_137,type,
sK59: ( real > nat ) > real ).
thf(func_def_138,type,
sK60: ( real > nat ) > real ).
thf(func_def_139,type,
sK61: ( nat > real ) > nat ).
thf(func_def_140,type,
sK62: ( nat > real ) > nat ).
thf(func_def_141,type,
sK63: ( nat > nat ) > nat ).
thf(func_def_142,type,
sK64: ( nat > nat ) > nat ).
thf(func_def_143,type,
sK65: ( real > nat ) > real ).
thf(func_def_144,type,
sK66: ( real > nat ) > real ).
thf(func_def_145,type,
sK67: ( real > real > $o ) > real ).
thf(func_def_146,type,
sK68: ( real > real > $o ) > real ).
thf(func_def_147,type,
sK69: ( real > real > $o ) > real ).
thf(func_def_148,type,
sK70: ( real > real > $o ) > real ).
thf(func_def_149,type,
sK71: ( real > real > $o ) > real ).
thf(func_def_150,type,
sK72: ( nat > nat ) > nat ).
thf(func_def_151,type,
sK73: ( nat > nat ) > nat ).
thf(func_def_152,type,
sK74: ( real > nat ) > real ).
thf(func_def_153,type,
sK75: ( real > nat ) > real ).
thf(func_def_154,type,
sK76: ( real > real ) > real ).
thf(func_def_155,type,
sK77: ( real > real ) > real ).
thf(func_def_156,type,
sK78: ( nat > real ) > nat ).
thf(func_def_157,type,
sK79: ( nat > real ) > nat ).
thf(func_def_158,type,
sK80: ( real > real ) > real ).
thf(func_def_159,type,
sK81: ( real > real ) > real ).
thf(func_def_160,type,
sK82: ( real > nat ) > real ).
thf(func_def_161,type,
sK83: ( real > nat ) > real ).
thf(func_def_162,type,
sK84: ( real > real > $o ) > real ).
thf(func_def_163,type,
sK85: ( real > real > $o ) > real ).
thf(func_def_164,type,
sK86: ( real > real > $o ) > real ).
thf(func_def_165,type,
sK87: ( real > real > $o ) > real ).
thf(func_def_166,type,
sK88: ( nat > real ) > nat ).
thf(func_def_167,type,
sK89: ( nat > real ) > nat ).
thf(func_def_168,type,
sK90: ( nat > nat > $o ) > nat ).
thf(func_def_169,type,
sK91: ( nat > nat > $o ) > nat ).
thf(func_def_170,type,
sK92: ( nat > nat > $o ) > nat ).
thf(func_def_171,type,
sK93: ( nat > nat > $o ) > nat ).
thf(func_def_172,type,
sK94: real > real > poly_real > real ).
thf(func_def_173,type,
sK95: real > real > poly_real > real ).
thf(func_def_174,type,
sK96: real > real > poly_real > real ).
thf(func_def_175,type,
sK97: real > real ).
thf(func_def_176,type,
sK98: nat > ( nat > $o ) > nat ).
thf(func_def_177,type,
sK99: nat > ( nat > $o ) > nat ).
thf(func_def_178,type,
sK100: ( nat > nat ) > nat ).
thf(func_def_179,type,
sK101: ( nat > nat ) > nat ).
thf(func_def_180,type,
sK102: ( real > real ) > real ).
thf(func_def_181,type,
sK103: ( real > real ) > real ).
thf(func_def_182,type,
sK104: poly_real > real > real ).
thf(func_def_183,type,
sK105: ( nat > real ) > nat ).
thf(func_def_184,type,
sK106: ( nat > real ) > nat ).
thf(func_def_185,type,
sK107: ( nat > nat ) > nat ).
thf(func_def_186,type,
sK108: ( nat > nat ) > nat ).
thf(func_def_187,type,
sK109: nat > nat ).
thf(func_def_188,type,
sK110: ( nat > real ) > nat ).
thf(func_def_189,type,
sK111: ( nat > real ) > nat ).
thf(func_def_190,type,
sK112: ( nat > real ) > nat ).
thf(func_def_191,type,
sK113: ( nat > real ) > nat ).
thf(func_def_192,type,
sK114: ( nat > $o ) > nat ).
thf(func_def_193,type,
sK115: real > real > poly_real > real ).
thf(func_def_194,type,
sK116: real > real > real ).
thf(func_def_195,type,
sK117: ( nat > $o ) > nat ).
thf(func_def_196,type,
sK118: ( nat > nat ) > nat ).
thf(func_def_197,type,
sK119: ( nat > nat ) > nat ).
thf(func_def_198,type,
sK120: ( real > nat ) > real ).
thf(func_def_199,type,
sK121: ( real > nat ) > real ).
thf(func_def_200,type,
sK122: ( real > nat ) > real ).
thf(func_def_201,type,
sK123: ( real > nat ) > real ).
thf(func_def_202,type,
sK124: real > real ).
thf(func_def_203,type,
sK125: ( nat > real ) > nat ).
thf(func_def_204,type,
sK126: ( nat > real ) > nat ).
thf(func_def_205,type,
sK127: ( nat > nat ) > nat ).
thf(func_def_206,type,
sK128: ( nat > nat ) > nat ).
thf(func_def_207,type,
sK129: ( real > nat ) > real ).
thf(func_def_208,type,
sK130: ( real > nat ) > real ).
thf(func_def_209,type,
sK131: real > poly_real > real > real ).
thf(func_def_210,type,
sK132: real > poly_real > real > real ).
thf(func_def_211,type,
sK133: real > poly_real > real > real ).
thf(func_def_212,type,
sK134: ( real > nat ) > real ).
thf(func_def_213,type,
sK135: ( real > nat ) > real ).
thf(func_def_214,type,
sK136: real ).
thf(func_def_216,type,
ph138:
!>[X0: $tType] : X0 ).
thf(f2107,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2105,f2009]) ).
thf(f2009,plain,
( ( ord_less_real @ lb1 @ sK136 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1987]) ).
thf(f1987,plain,
( ( ( ord_less_real @ lb1 @ sK136 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1853,f1767]) ).
thf(f1767,plain,
! [X2: real,X0: real,X1: real] :
( ( ( ord_less_real @ ( ord_min_real @ X1 @ X2 ) @ X0 )
= $true )
| ( ( ord_less_real @ X1 @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1191]) ).
thf(f1191,plain,
! [X0: real,X1: real,X2: real] :
( ( ( ord_less_real @ X1 @ X0 )
!= $true )
| ( ( ord_less_real @ ( ord_min_real @ X1 @ X2 ) @ X0 )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f955]) ).
thf(f955,plain,
! [X0: real,X1: real,X2: real] :
( ( ( ord_less_real @ X1 @ X0 )
= $true )
=> ( ( ord_less_real @ ( ord_min_real @ X1 @ X2 ) @ X0 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f954]) ).
thf(f954,plain,
! [X0: real,X1: real,X2: real] :
( ( ord_less_real @ X1 @ X0 )
=> ( ord_less_real @ ( ord_min_real @ X1 @ X2 ) @ X0 ) ),
inference(rectify,[status(thm)],[f42]) ).
thf(f42,axiom,
! [X16: real,X4: real,X5: real] :
( ( ord_less_real @ X4 @ X16 )
=> ( ord_less_real @ ( ord_min_real @ X4 @ X5 ) @ X16 ) ),
file('/export/starexec/sandbox/tmp/tmp.iTAcLxWr78/DTF2THF_18417.p',fact_41_min_Ostrict__coboundedI1) ).
thf(f1853,plain,
( ( ord_less_real @ ( ord_min_real @ lb1 @ lb2 ) @ sK136 )
!= $true ),
inference(definition_unfolding,[status(thm)],[f1813,f1530]) ).
thf(f1530,plain,
( lb
= ( ord_min_real @ lb1 @ lb2 ) ),
inference(cnf_transformation,[status(thm)],[f2]) ).
thf(f2,axiom,
( lb
= ( ord_min_real @ lb1 @ lb2 ) ),
file('/export/starexec/sandbox/tmp/tmp.iTAcLxWr78/DTF2THF_18417.p',fact_1_lb__def) ).
thf(f1813,plain,
( ( ord_less_real @ lb @ sK136 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1464]) ).
thf(f1464,plain,
( ( zero_zero_real
= ( poly_real2 @ p @ sK136 ) )
& ( ( ord_less_real @ lb @ sK136 )
!= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK136])],[f1209,f1463]) ).
thf(f1463,plain,
( ? [X0: real] :
( ( ( poly_real2 @ p @ X0 )
= zero_zero_real )
& ( ( ord_less_real @ lb @ X0 )
!= $true ) )
=> ( ( zero_zero_real
= ( poly_real2 @ p @ sK136 ) )
& ( ( ord_less_real @ lb @ sK136 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1209,plain,
? [X0: real] :
( ( ( poly_real2 @ p @ X0 )
= zero_zero_real )
& ( ( ord_less_real @ lb @ X0 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f451]) ).
thf(f451,plain,
~ ! [X0: real] :
( ( ( poly_real2 @ p @ X0 )
!= zero_zero_real )
| ( ( ord_less_real @ lb @ X0 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f450]) ).
thf(f450,plain,
~ ! [X0: real] :
( ( ( poly_real2 @ p @ X0 )
!= zero_zero_real )
| ( ord_less_real @ lb @ X0 ) ),
inference(rectify,[status(thm)],[f352]) ).
thf(f352,negated_conjecture,
~ ! [X7: real] :
( ( zero_zero_real
!= ( poly_real2 @ p @ X7 ) )
| ( ord_less_real @ lb @ X7 ) ),
inference(negated_conjecture,[status(cth)],[f351]) ).
thf(f351,conjecture,
! [X7: real] :
( ( zero_zero_real
!= ( poly_real2 @ p @ X7 ) )
| ( ord_less_real @ lb @ X7 ) ),
file('/export/starexec/sandbox/tmp/tmp.iTAcLxWr78/DTF2THF_18417.p',conj_0) ).
thf(f2105,plain,
( ( ord_less_real @ lb1 @ sK136 )
= $true ),
inference(equality_resolution,[status(thm)],[f1837]) ).
thf(f1837,plain,
! [X0: real] :
( ( ( poly_real2 @ p @ X0 )
!= ( poly_real2 @ p @ sK136 ) )
| ( ( ord_less_real @ lb1 @ X0 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f1752,f1814]) ).
thf(f1814,plain,
( zero_zero_real
= ( poly_real2 @ p @ sK136 ) ),
inference(cnf_transformation,[status(thm)],[f1464]) ).
thf(f1752,plain,
! [X0: real] :
( ( ( ord_less_real @ lb1 @ X0 )
= $true )
| ( ( poly_real2 @ p @ X0 )
!= zero_zero_real ) ),
inference(cnf_transformation,[status(thm)],[f1054]) ).
thf(f1054,plain,
! [X0: real] :
( ( ( ord_less_real @ lb1 @ X0 )
= $true )
| ( ( poly_real2 @ p @ X0 )
!= zero_zero_real ) ),
inference(ennf_transformation,[status(thm)],[f447]) ).
thf(f447,plain,
! [X0: real] :
( ( ( poly_real2 @ p @ X0 )
= zero_zero_real )
=> ( ( ord_less_real @ lb1 @ X0 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f446]) ).
thf(f446,plain,
! [X0: real] :
( ( ( poly_real2 @ p @ X0 )
= zero_zero_real )
=> ( ord_less_real @ lb1 @ X0 ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: real] :
( ( ( poly_real2 @ p @ X0 )
= zero_zero_real )
=> ( ord_less_real @ lb1 @ X0 ) ),
file('/export/starexec/sandbox/tmp/tmp.iTAcLxWr78/DTF2THF_18417.p',fact_0_lb1) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP194^1 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n018.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:29:31 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.47 thf(spec,logic,$$dhol).
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % File : ITP194^1 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.47 % Domain : Interactive Theorem Proving
% 0.20/0.47 % Problem : Sledgehammer Sturm_Tarski problem prob_257__5870986_1
% 0.20/0.47 % Version : Especial.
% 0.20/0.47 % English :
% 0.20/0.47
% 0.20/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.47 % Source : [Des21]
% 0.20/0.47 % Names : Sturm_Tarski/prob_257__5870986_1 [Des21]
% 0.20/0.47
% 0.20/0.47 % Status : Theorem
% 0.20/0.47 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0
% 0.20/0.47 % Syntax : Number of formulae : 415 ( 169 unt; 64 typ; 0 def)
% 0.20/0.47 % Number of atoms : 985 ( 326 equ; 0 cnn)
% 0.20/0.47 % Maximal formula atoms : 12 ( 2 avg)
% 0.20/0.47 % Number of connectives : 2504 ( 135 ~; 28 |; 37 &;1867 @)
% 0.20/0.47 % ( 0 <=>; 437 =>; 0 <=; 0 <~>)
% 0.20/0.47 % Maximal formula depth : 14 ( 6 avg)
% 0.20/0.47 % Number of types : 9 ( 8 usr)
% 0.20/0.47 % Number of type conns : 151 ( 151 >; 0 *; 0 +; 0 <<)
% 0.20/0.47 % Number of symbols : 59 ( 56 usr; 14 con; 0-6 aty)
% 0.20/0.47 % Number of variables : 907 ( 68 ^; 823 !; 16 ?; 907 :)
% 0.20/0.47 % SPC : TH0_THM_EQU_NAR
% 0.20/0.47
% 0.20/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:41:56.214
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % Could-be-implicit typings (8)
% 0.20/0.47 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 0.20/0.47 poly_poly_poly_real: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.20/0.47 poly_poly_real: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 0.20/0.47 poly_poly_nat: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 poly_real: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.20/0.47 poly_nat: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.20/0.47 set_real: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_n_t__Real__Oreal,type,
% 0.20/0.47 real: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.20/0.47 nat: $tType ).
% 0.20/0.47
% 0.20/0.47 % Explicit typings (56)
% 0.20/0.47 thf(sy_c_Groups_Osgn__class_Osgn_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.20/0.47 sgn_sg2128174761y_real: poly_poly_real > poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Osgn__class_Osgn_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 sgn_sgn_poly_real: poly_real > poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
% 0.20/0.47 sgn_sgn_real: real > real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.20/0.47 zero_zero_nat: nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.20/0.47 zero_zero_poly_nat: poly_nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 0.20/0.47 zero_z1059985641ly_nat: poly_poly_nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 0.20/0.47 zero_z935034829y_real: poly_poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.20/0.47 zero_z1423781445y_real: poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 zero_zero_poly_real: poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.20/0.47 zero_zero_real: real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_If_001t__Nat__Onat,type,
% 0.20/0.47 if_nat: $o > nat > nat > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_If_001t__Real__Oreal,type,
% 0.20/0.47 if_real: $o > real > real > real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.20/0.47 ord_less_nat: nat > nat > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.20/0.47 ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 ord_less_poly_real: poly_real > poly_real > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.20/0.47 ord_less_real: real > real > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.20/0.47 ord_less_eq_nat: nat > nat > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.20/0.47 ord_le893774876y_real: poly_poly_real > poly_poly_real > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 ord_le1180086932y_real: poly_real > poly_real > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.20/0.47 ord_less_eq_real: real > real > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
% 0.20/0.47 ord_min_nat: nat > nat > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Omin_001t__Real__Oreal,type,
% 0.20/0.47 ord_min_real: real > real > real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.20/0.47 divide924636027y_real: poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > nat > nat > poly_poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Odivide__poly__main_001t__Real__Oreal,type,
% 0.20/0.47 divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat,type,
% 0.20/0.47 is_zero_nat: poly_nat > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 is_zero_poly_real: poly_poly_real > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Ois__zero_001t__Real__Oreal,type,
% 0.20/0.47 is_zero_real: poly_real > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.20/0.47 order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 order_poly_real: poly_real > poly_poly_real > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oorder_001t__Real__Oreal,type,
% 0.20/0.47 order_real: real > poly_real > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
% 0.20/0.47 poly_nat2: poly_nat > nat > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.20/0.47 poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.20/0.47 poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 poly_poly_real2: poly_poly_real > poly_real > poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
% 0.20/0.47 poly_real2: poly_real > real > real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat,type,
% 0.20/0.47 poly_cutoff_nat: nat > poly_nat > poly_nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly__cutoff_001t__Real__Oreal,type,
% 0.20/0.47 poly_cutoff_real: nat > poly_real > poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat,type,
% 0.20/0.47 poly_shift_nat: nat > poly_nat > poly_nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly__shift_001t__Real__Oreal,type,
% 0.20/0.47 poly_shift_real: nat > poly_real > poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
% 0.20/0.47 reflect_poly_nat: poly_nat > poly_nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.20/0.47 reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.20/0.47 reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.20/0.47 reflec1522834046y_real: poly_poly_real > poly_poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal,type,
% 0.20/0.47 reflect_poly_real: poly_real > poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.20/0.47 collect_real: ( real > $o ) > set_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Osgn__neg__inf_001t__Real__Oreal,type,
% 0.20/0.47 sturm_1076696862f_real: poly_real > real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Osgn__pos__inf_001t__Real__Oreal,type,
% 0.20/0.47 sturm_1308388506f_real: poly_real > real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_member_001t__Real__Oreal,type,
% 0.20/0.47 member_real: real > set_real > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_lb1____,type,
% 0.20/0.47 lb1: real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_lb2____,type,
% 0.20/0.47 lb2: real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_lb____,type,
% 0.20/0.47 lb: real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_p,type,
% 0.20/0.47 p: poly_real ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_thesis,type,
% 0.20/0.47 thesis: $o ).
% 0.20/0.47
% 0.20/0.47 % Relevant facts (345)
% 0.20/0.47 thf(fact_0_lb1,axiom,
% 0.20/0.47 ! [X: real] :
% 0.20/0.47 ( ( ( poly_real2 @ p @ X )
% 0.20/0.47 = zero_zero_real )
% 0.20/0.47 => ( ord_less_real @ lb1 @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % lb1
% 0.20/0.47 thf(fact_1_lb__def,axiom,
% 0.20/0.47 ( lb
% 0.20/0.47 = ( ord_min_real @ lb1 @ lb2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % lb_def
% 0.20/0.47 thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb1_O_A_092_060forall_062x_O_Apoly_Ap_Ax_A_061_A0_A_092_060longrightarrow_062_Alb1_A_060_Ax_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.20/0.47 ~ ! [Lb1: real] :
% 0.20/0.47 ~ ! [X: real] :
% 0.20/0.47 ( ( ( poly_real2 @ p @ X )
% 0.20/0.47 = zero_zero_real )
% 0.20/0.47 => ( ord_less_real @ Lb1 @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % \<open>\<And>thesis. (\<And>lb1. \<forall>x. poly p x = 0 \<longrightarrow> lb1 < x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.20/0.47 thf(fact_3_assms,axiom,
% 0.20/0.47 p != zero_zero_poly_real ).
% 0.20/0.47
% 0.20/0.47 % assms
% 0.20/0.47 thf(fact_4_poly__0,axiom,
% 0.20/0.47 ! [X2: poly_nat] :
% 0.20/0.47 ( ( poly_poly_nat2 @ zero_z1059985641ly_nat @ X2 )
% 0.20/0.47 = zero_zero_poly_nat ) ).
% 0.20/0.47
% 0.20/0.47 % poly_0
% 0.20/0.47 thf(fact_5_poly__0,axiom,
% 0.20/0.47 ! [X2: poly_poly_real] :
% 0.20/0.47 ( ( poly_poly_poly_real2 @ zero_z935034829y_real @ X2 )
% 0.20/0.47 = zero_z1423781445y_real ) ).
% 0.20/0.47
% 0.20/0.47 % poly_0
% 0.20/0.47 thf(fact_6_poly__0,axiom,
% 0.20/0.47 ! [X2: poly_real] :
% 0.20/0.47 ( ( poly_poly_real2 @ zero_z1423781445y_real @ X2 )
% 0.20/0.47 = zero_zero_poly_real ) ).
% 0.20/0.47
% 0.20/0.47 % poly_0
% 0.20/0.47 thf(fact_7_poly__0,axiom,
% 0.20/0.47 ! [X2: nat] :
% 0.20/0.47 ( ( poly_nat2 @ zero_zero_poly_nat @ X2 )
% 0.20/0.47 = zero_zero_nat ) ).
% 0.20/0.47
% 0.20/0.47 % poly_0
% 0.20/0.47 thf(fact_8_poly__0,axiom,
% 0.20/0.47 ! [X2: real] :
% 0.20/0.47 ( ( poly_real2 @ zero_zero_poly_real @ X2 )
% 0.20/0.47 = zero_zero_real ) ).
% 0.20/0.47
% 0.20/0.47 % poly_0
% 0.20/0.47 thf(fact_9_not__gr__zero,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.20/0.47 = ( N = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_gr_zero
% 0.20/0.47 thf(fact_10_poly__IVT__neg,axiom,
% 0.20/0.47 ! [A: real,B: real,P: poly_real] :
% 0.20/0.47 ( ( ord_less_real @ A @ B )
% 0.20/0.47 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A ) )
% 0.20/0.47 => ( ( ord_less_real @ ( poly_real2 @ P @ B ) @ zero_zero_real )
% 0.20/0.47 => ? [X3: real] :
% 0.20/0.47 ( ( ord_less_real @ A @ X3 )
% 0.20/0.47 & ( ord_less_real @ X3 @ B )
% 0.20/0.47 & ( ( poly_real2 @ P @ X3 )
% 0.20/0.47 = zero_zero_real ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % poly_IVT_neg
% 0.20/0.47 thf(fact_11_poly__IVT__pos,axiom,
% 0.20/0.47 ! [A: real,B: real,P: poly_real] :
% 0.20/0.47 ( ( ord_less_real @ A @ B )
% 0.20/0.47 => ( ( ord_less_real @ ( poly_real2 @ P @ A ) @ zero_zero_real )
% 0.20/0.47 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B ) )
% 0.20/0.47 => ? [X3: real] :
% 0.20/0.47 ( ( ord_less_real @ A @ X3 )
% 0.20/0.47 & ( ord_less_real @ X3 @ B )
% 0.20/0.47 & ( ( poly_real2 @ P @ X3 )
% 0.20/0.47 = zero_zero_real ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % poly_IVT_pos
% 0.20/0.47 thf(fact_12_poly__all__0__iff__0,axiom,
% 0.20/0.47 ! [P: poly_poly_poly_real] :
% 0.20/0.47 ( ( ! [X4: poly_poly_real] :
% 0.20/0.47 ( ( poly_poly_poly_real2 @ P @ X4 )
% 0.20/0.47 = zero_z1423781445y_real ) )
% 0.20/0.47 = ( P = zero_z935034829y_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % poly_all_0_iff_0
% 0.20/0.47 thf(fact_13_poly__all__0__iff__0,axiom,
% 0.20/0.47 ! [P: poly_real] :
% 0.20/0.47 ( ( ! [X4: real] :
% 0.20/0.47 ( ( poly_real2 @ P @ X4 )
% 0.20/0.47 = zero_zero_real ) )
% 0.20/0.47 = ( P = zero_zero_poly_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % poly_all_0_iff_0
% 0.20/0.47 thf(fact_14_poly__all__0__iff__0,axiom,
% 0.20/0.47 ! [P: poly_poly_real] :
% 0.20/0.47 ( ( ! [X4: poly_real] :
% 0.20/0.47 ( ( poly_poly_real2 @ P @ X4 )
% 0.20/0.47 = zero_zero_poly_real ) )
% 0.20/0.47 = ( P = zero_z1423781445y_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % poly_all_0_iff_0
% 0.20/0.47 thf(fact_15_less__numeral__extra_I3_J,axiom,
% 0.20/0.47 ~ ( ord_less_poly_real @ zero_zero_poly_real @ zero_zero_poly_real ) ).
% 0.20/0.47
% 0.20/0.47 % less_numeral_extra(3)
% 0.20/0.47 thf(fact_16_less__numeral__extra_I3_J,axiom,
% 0.20/0.47 ~ ( ord_le38482960y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ) ).
% 0.20/0.47
% 0.20/0.47 % less_numeral_extra(3)
% 0.20/0.47 thf(fact_17_less__numeral__extra_I3_J,axiom,
% 0.20/0.47 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.20/0.47
% 0.20/0.47 % less_numeral_extra(3)
% 0.20/0.47 thf(fact_18_less__numeral__extra_I3_J,axiom,
% 0.20/0.47 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.20/0.47
% 0.20/0.47 % less_numeral_extra(3)
% 0.20/0.47 thf(fact_19_field__lbound__gt__zero,axiom,
% 0.20/0.47 ! [D1: real,D2: real] :
% 0.20/0.47 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.20/0.47 => ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.20/0.47 => ? [E: real] :
% 0.20/0.47 ( ( ord_less_real @ zero_zero_real @ E )
% 0.20/0.47 & ( ord_less_real @ E @ D1 )
% 0.20/0.47 & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % field_lbound_gt_zero
% 0.20/0.47 thf(fact_20_gr__zeroI,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( N != zero_zero_nat )
% 0.20/0.47 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % gr_zeroI
% 0.20/0.47 thf(fact_21_not__less__zero,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.20/0.47
% 0.20/0.47 % not_less_zero
% 0.20/0.47 thf(fact_22_gr__implies__not__zero,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less_nat @ M @ N )
% 0.20/0.47 => ( N != zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % gr_implies_not_zero
% 0.20/0.47 thf(fact_23_zero__less__iff__neq__zero,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.20/0.47 = ( N != zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_less_iff_neq_zero
% 0.20/0.47 thf(fact_24_zero__reorient,axiom,
% 0.20/0.47 ! [X2: real] :
% 0.20/0.47 ( ( zero_zero_real = X2 )
% 0.20/0.47 = ( X2 = zero_zero_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_reorient
% 0.20/0.47 thf(fact_25_zero__reorient,axiom,
% 0.20/0.47 ! [X2: poly_real] :
% 0.20/0.47 ( ( zero_zero_poly_real = X2 )
% 0.20/0.47 = ( X2 = zero_zero_poly_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_reorient
% 0.20/0.47 thf(fact_26_zero__reorient,axiom,
% 0.20/0.47 ! [X2: nat] :
% 0.20/0.47 ( ( zero_zero_nat = X2 )
% 0.20/0.47 = ( X2 = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_reorient
% 0.20/0.47 thf(fact_27_zero__reorient,axiom,
% 0.20/0.47 ! [X2: poly_nat] :
% 0.20/0.47 ( ( zero_zero_poly_nat = X2 )
% 0.20/0.47 = ( X2 = zero_zero_poly_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_reorient
% 0.20/0.47 thf(fact_28_zero__reorient,axiom,
% 0.20/0.47 ! [X2: poly_poly_real] :
% 0.20/0.47 ( ( zero_z1423781445y_real = X2 )
% 0.20/0.47 = ( X2 = zero_z1423781445y_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_reorient
% 0.20/0.47 thf(fact_29_poly__eq__poly__eq__iff,axiom,
% 0.20/0.47 ! [P: poly_real,Q: poly_real] :
% 0.20/0.47 ( ( ( poly_real2 @ P )
% 0.20/0.47 = ( poly_real2 @ Q ) )
% 0.20/0.47 = ( P = Q ) ) ).
% 0.20/0.47
% 0.20/0.47 % poly_eq_poly_eq_iff
% 0.20/0.47 thf(fact_30_poly__eq__poly__eq__iff,axiom,
% 0.20/0.47 ! [P: poly_poly_real,Q: poly_poly_real] :
% 0.20/0.47 ( ( ( poly_poly_real2 @ P )
% 0.20/0.47 = ( poly_poly_real2 @ Q ) )
% 0.20/0.47 = ( P = Q ) ) ).
% 0.20/0.47
% 0.20/0.47 % poly_eq_poly_eq_iff
% 0.20/0.47 thf(fact_31_min__less__iff__conj,axiom,
% 0.20/0.47 ! [Z: real,X2: real,Y: real] :
% 0.20/0.47 ( ( ord_less_real @ Z @ ( ord_min_real @ X2 @ Y ) )
% 0.20/0.47 = ( ( ord_less_real @ Z @ X2 )
% 0.20/0.47 & ( ord_less_real @ Z @ Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_less_iff_conj
% 0.20/0.47 thf(fact_32_min__less__iff__conj,axiom,
% 0.20/0.47 ! [Z: nat,X2: nat,Y: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z @ ( ord_min_nat @ X2 @ Y ) )
% 0.20/0.47 = ( ( ord_less_nat @ Z @ X2 )
% 0.20/0.47 & ( ord_less_nat @ Z @ Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_less_iff_conj
% 0.20/0.47 thf(fact_33_min_Oidem,axiom,
% 0.20/0.47 ! [A: real] :
% 0.20/0.47 ( ( ord_min_real @ A @ A )
% 0.20/0.47 = A ) ).
% 0.20/0.47
% 0.20/0.47 % min.idem
% 0.20/0.47 thf(fact_34_min_Oidem,axiom,
% 0.20/0.47 ! [A: nat] :
% 0.20/0.47 ( ( ord_min_nat @ A @ A )
% 0.20/0.47 = A ) ).
% 0.20/0.47
% 0.20/0.47 % min.idem
% 0.20/0.47 thf(fact_35_min_Oleft__idem,axiom,
% 0.20/0.47 ! [A: real,B: real] :
% 0.20/0.47 ( ( ord_min_real @ A @ ( ord_min_real @ A @ B ) )
% 0.20/0.47 = ( ord_min_real @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.left_idem
% 0.20/0.47 thf(fact_36_min_Oleft__idem,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
% 0.20/0.47 = ( ord_min_nat @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.left_idem
% 0.20/0.47 thf(fact_37_min_Oright__idem,axiom,
% 0.20/0.47 ! [A: real,B: real] :
% 0.20/0.47 ( ( ord_min_real @ ( ord_min_real @ A @ B ) @ B )
% 0.20/0.47 = ( ord_min_real @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.right_idem
% 0.20/0.47 thf(fact_38_min_Oright__idem,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
% 0.20/0.47 = ( ord_min_nat @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.right_idem
% 0.20/0.47 thf(fact_39_min_Ostrict__coboundedI2,axiom,
% 0.20/0.47 ! [B: real,C: real,A: real] :
% 0.20/0.47 ( ( ord_less_real @ B @ C )
% 0.20/0.47 => ( ord_less_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.strict_coboundedI2
% 0.20/0.47 thf(fact_40_min_Ostrict__coboundedI2,axiom,
% 0.20/0.47 ! [B: nat,C: nat,A: nat] :
% 0.20/0.47 ( ( ord_less_nat @ B @ C )
% 0.20/0.47 => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.strict_coboundedI2
% 0.20/0.47 thf(fact_41_min_Ostrict__coboundedI1,axiom,
% 0.20/0.47 ! [A: real,C: real,B: real] :
% 0.20/0.47 ( ( ord_less_real @ A @ C )
% 0.20/0.47 => ( ord_less_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.strict_coboundedI1
% 0.20/0.47 thf(fact_42_min_Ostrict__coboundedI1,axiom,
% 0.20/0.47 ! [A: nat,C: nat,B: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ C )
% 0.20/0.47 => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.strict_coboundedI1
% 0.20/0.47 thf(fact_43_min_Ostrict__order__iff,axiom,
% 0.20/0.47 ( ord_less_real
% 0.20/0.47 = ( ^ [A2: real,B2: real] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 = ( ord_min_real @ A2 @ B2 ) )
% 0.20/0.47 & ( A2 != B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.strict_order_iff
% 0.20/0.47 thf(fact_44_min_Ostrict__order__iff,axiom,
% 0.20/0.47 ( ord_less_nat
% 0.20/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 = ( ord_min_nat @ A2 @ B2 ) )
% 0.20/0.47 & ( A2 != B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.strict_order_iff
% 0.20/0.47 thf(fact_45_min_Ostrict__boundedE,axiom,
% 0.20/0.47 ! [A: real,B: real,C: real] :
% 0.20/0.47 ( ( ord_less_real @ A @ ( ord_min_real @ B @ C ) )
% 0.20/0.47 => ~ ( ( ord_less_real @ A @ B )
% 0.20/0.47 => ~ ( ord_less_real @ A @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.strict_boundedE
% 0.20/0.47 thf(fact_46_min_Ostrict__boundedE,axiom,
% 0.20/0.47 ! [A: nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
% 0.20/0.47 => ~ ( ( ord_less_nat @ A @ B )
% 0.20/0.47 => ~ ( ord_less_nat @ A @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.strict_boundedE
% 0.20/0.47 thf(fact_47_min__less__iff__disj,axiom,
% 0.20/0.47 ! [X2: real,Y: real,Z: real] :
% 0.20/0.47 ( ( ord_less_real @ ( ord_min_real @ X2 @ Y ) @ Z )
% 0.20/0.47 = ( ( ord_less_real @ X2 @ Z )
% 0.20/0.47 | ( ord_less_real @ Y @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_less_iff_disj
% 0.20/0.47 thf(fact_48_min__less__iff__disj,axiom,
% 0.20/0.47 ! [X2: nat,Y: nat,Z: nat] :
% 0.20/0.47 ( ( ord_less_nat @ ( ord_min_nat @ X2 @ Y ) @ Z )
% 0.20/0.47 = ( ( ord_less_nat @ X2 @ Z )
% 0.20/0.47 | ( ord_less_nat @ Y @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_less_iff_disj
% 0.20/0.47 thf(fact_49_is__zero__null,axiom,
% 0.20/0.47 ( is_zero_real
% 0.20/0.47 = ( ^ [P2: poly_real] : ( P2 = zero_zero_poly_real ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_zero_null
% 0.20/0.47 thf(fact_50_is__zero__null,axiom,
% 0.20/0.47 ( is_zero_nat
% 0.20/0.47 = ( ^ [P2: poly_nat] : ( P2 = zero_zero_poly_nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_zero_null
% 0.20/0.47 thf(fact_51_is__zero__null,axiom,
% 0.20/0.47 ( is_zero_poly_real
% 0.20/0.47 = ( ^ [P2: poly_poly_real] : ( P2 = zero_z1423781445y_real ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_zero_null
% 0.20/0.47 thf(fact_52_min_Oleft__commute,axiom,
% 0.20/0.47 ! [B: real,A: real,C: real] :
% 0.20/0.47 ( ( ord_min_real @ B @ ( ord_min_real @ A @ C ) )
% 0.20/0.47 = ( ord_min_real @ A @ ( ord_min_real @ B @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.left_commute
% 0.20/0.47 thf(fact_53_min_Oleft__commute,axiom,
% 0.20/0.47 ! [B: nat,A: nat,C: nat] :
% 0.20/0.47 ( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
% 0.20/0.47 = ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.left_commute
% 0.20/0.47 thf(fact_54_min_Ocommute,axiom,
% 0.20/0.47 ( ord_min_real
% 0.20/0.47 = ( ^ [A2: real,B2: real] : ( ord_min_real @ B2 @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.commute
% 0.20/0.47 thf(fact_55_min_Ocommute,axiom,
% 0.20/0.47 ( ord_min_nat
% 0.20/0.47 = ( ^ [A2: nat,B2: nat] : ( ord_min_nat @ B2 @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.commute
% 0.20/0.47 thf(fact_56_min_Oassoc,axiom,
% 0.20/0.47 ! [A: real,B: real,C: real] :
% 0.20/0.47 ( ( ord_min_real @ ( ord_min_real @ A @ B ) @ C )
% 0.20/0.47 = ( ord_min_real @ A @ ( ord_min_real @ B @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.assoc
% 0.20/0.47 thf(fact_57_min_Oassoc,axiom,
% 0.20/0.47 ! [A: nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
% 0.20/0.47 = ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.assoc
% 0.20/0.47 thf(fact_58_poly__cutoff__0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( poly_cutoff_real @ N @ zero_zero_poly_real )
% 0.20/0.47 = zero_zero_poly_real ) ).
% 0.20/0.47
% 0.20/0.47 % poly_cutoff_0
% 0.20/0.47 thf(fact_59_poly__cutoff__0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
% 0.20/0.47 = zero_zero_poly_nat ) ).
% 0.20/0.47
% 0.20/0.47 % poly_cutoff_0
% 0.20/0.47 thf(fact_60_poly__cutoff__0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( poly_c1404107022y_real @ N @ zero_z1423781445y_real )
% 0.20/0.47 = zero_z1423781445y_real ) ).
% 0.20/0.47
% 0.20/0.47 % poly_cutoff_0
% 0.20/0.47 thf(fact_61_last__non__root__interval,axiom,
% 0.20/0.47 ! [P: poly_real,Ub: real] :
% 0.20/0.47 ( ( P != zero_zero_poly_real )
% 0.20/0.47 => ~ ! [Lb: real] :
% 0.20/0.47 ( ( ord_less_real @ Lb @ Ub )
% 0.20/0.47 => ~ ! [Z2: real] :
% 0.20/0.47 ( ( ( ord_less_eq_real @ Lb @ Z2 )
% 0.20/0.47 & ( ord_less_real @ Z2 @ Ub ) )
% 0.20/0.47 => ( ( poly_real2 @ P @ Z2 )
% 0.20/0.47 != zero_zero_real ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % last_non_root_interval
% 0.20/0.47 thf(fact_62_next__non__root__interval,axiom,
% 0.20/0.47 ! [P: poly_real,Lb2: real] :
% 0.20/0.47 ( ( P != zero_zero_poly_real )
% 0.20/0.47 => ~ ! [Ub2: real] :
% 0.20/0.47 ( ( ord_less_real @ Lb2 @ Ub2 )
% 0.20/0.47 => ~ ! [Z2: real] :
% 0.20/0.47 ( ( ( ord_less_real @ Lb2 @ Z2 )
% 0.20/0.47 & ( ord_less_eq_real @ Z2 @ Ub2 ) )
% 0.20/0.47 => ( ( poly_real2 @ P @ Z2 )
% 0.20/0.47 != zero_zero_real ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % next_non_root_interval
% 0.20/0.47 thf(fact_63_reflect__poly__at__0__eq__0__iff,axiom,
% 0.20/0.47 ! [P: poly_real] :
% 0.20/0.47 ( ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
% 0.20/0.47 = zero_zero_real )
% 0.20/0.47 = ( P = zero_zero_poly_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflect_poly_at_0_eq_0_iff
% 0.20/0.47 thf(fact_64_reflect__poly__at__0__eq__0__iff,axiom,
% 0.20/0.47 ! [P: poly_poly_real] :
% 0.20/0.47 ( ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
% 0.20/0.47 = zero_zero_poly_real )
% 0.20/0.47 = ( P = zero_z1423781445y_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflect_poly_at_0_eq_0_iff
% 0.20/0.47 thf(fact_65_reflect__poly__at__0__eq__0__iff,axiom,
% 0.20/0.47 ! [P: poly_nat] :
% 0.20/0.47 ( ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 0.20/0.47 = zero_zero_nat )
% 0.20/0.47 = ( P = zero_zero_poly_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflect_poly_at_0_eq_0_iff
% 0.20/0.47 thf(fact_66_reflect__poly__at__0__eq__0__iff,axiom,
% 0.20/0.47 ! [P: poly_poly_nat] :
% 0.20/0.47 ( ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
% 0.20/0.47 = zero_zero_poly_nat )
% 0.20/0.47 = ( P = zero_z1059985641ly_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflect_poly_at_0_eq_0_iff
% 0.20/0.47 thf(fact_67_reflect__poly__at__0__eq__0__iff,axiom,
% 0.20/0.47 ! [P: poly_poly_poly_real] :
% 0.20/0.47 ( ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
% 0.20/0.47 = zero_z1423781445y_real )
% 0.20/0.47 = ( P = zero_z935034829y_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflect_poly_at_0_eq_0_iff
% 0.20/0.47 thf(fact_68_lb2,axiom,
% 0.20/0.47 ! [X: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ X @ lb2 )
% 0.20/0.47 => ( ( sgn_sgn_real @ ( poly_real2 @ p @ X ) )
% 0.20/0.47 = ( sturm_1076696862f_real @ p ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % lb2
% 0.20/0.47 thf(fact_69_poly__shift__0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( poly_shift_real @ N @ zero_zero_poly_real )
% 0.20/0.47 = zero_zero_poly_real ) ).
% 0.20/0.47
% 0.20/0.47 % poly_shift_0
% 0.20/0.47 thf(fact_70_poly__shift__0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( poly_shift_nat @ N @ zero_zero_poly_nat )
% 0.20/0.47 = zero_zero_poly_nat ) ).
% 0.20/0.47
% 0.20/0.47 % poly_shift_0
% 0.20/0.47 thf(fact_71_poly__shift__0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( poly_shift_poly_real @ N @ zero_z1423781445y_real )
% 0.20/0.47 = zero_z1423781445y_real ) ).
% 0.20/0.47
% 0.20/0.47 % poly_shift_0
% 0.20/0.47 thf(fact_72_order__root,axiom,
% 0.20/0.47 ! [P: poly_real,A: real] :
% 0.20/0.47 ( ( ( poly_real2 @ P @ A )
% 0.20/0.47 = zero_zero_real )
% 0.20/0.47 = ( ( P = zero_zero_poly_real )
% 0.20/0.47 | ( ( order_real @ A @ P )
% 0.20/0.47 != zero_zero_nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_root
% 0.20/0.47 thf(fact_73_order__root,axiom,
% 0.20/0.47 ! [P: poly_poly_real,A: poly_real] :
% 0.20/0.47 ( ( ( poly_poly_real2 @ P @ A )
% 0.20/0.47 = zero_zero_poly_real )
% 0.20/0.47 = ( ( P = zero_z1423781445y_real )
% 0.20/0.47 | ( ( order_poly_real @ A @ P )
% 0.20/0.47 != zero_zero_nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_root
% 0.20/0.47 thf(fact_74_order__root,axiom,
% 0.20/0.47 ! [P: poly_poly_poly_real,A: poly_poly_real] :
% 0.20/0.47 ( ( ( poly_poly_poly_real2 @ P @ A )
% 0.20/0.47 = zero_z1423781445y_real )
% 0.20/0.47 = ( ( P = zero_z935034829y_real )
% 0.20/0.47 | ( ( order_poly_poly_real @ A @ P )
% 0.20/0.47 != zero_zero_nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_root
% 0.20/0.47 thf(fact_75_divide__poly__main__0,axiom,
% 0.20/0.47 ! [R: poly_real,D: poly_real,Dr: nat,N: nat] :
% 0.20/0.47 ( ( divide1561404011n_real @ zero_zero_real @ zero_zero_poly_real @ R @ D @ Dr @ N )
% 0.20/0.47 = zero_zero_poly_real ) ).
% 0.20/0.47
% 0.20/0.47 % divide_poly_main_0
% 0.20/0.47 thf(fact_76_divide__poly__main__0,axiom,
% 0.20/0.47 ! [R: poly_poly_real,D: poly_poly_real,Dr: nat,N: nat] :
% 0.20/0.47 ( ( divide1142363123y_real @ zero_zero_poly_real @ zero_z1423781445y_real @ R @ D @ Dr @ N )
% 0.20/0.47 = zero_z1423781445y_real ) ).
% 0.20/0.47
% 0.20/0.47 % divide_poly_main_0
% 0.20/0.47 thf(fact_77_divide__poly__main__0,axiom,
% 0.20/0.47 ! [R: poly_poly_poly_real,D: poly_poly_poly_real,Dr: nat,N: nat] :
% 0.20/0.47 ( ( divide924636027y_real @ zero_z1423781445y_real @ zero_z935034829y_real @ R @ D @ Dr @ N )
% 0.20/0.47 = zero_z935034829y_real ) ).
% 0.20/0.47
% 0.20/0.47 % divide_poly_main_0
% 0.20/0.47 thf(fact_78_not__eq__pos__or__neg__iff__1,axiom,
% 0.20/0.47 ! [Lb2: real,Ub: real,P: poly_real] :
% 0.20/0.47 ( ( ! [Z3: real] :
% 0.20/0.47 ( ( ( ord_less_real @ Lb2 @ Z3 )
% 0.20/0.47 & ( ord_less_eq_real @ Z3 @ Ub ) )
% 0.20/0.47 => ( ( poly_real2 @ P @ Z3 )
% 0.20/0.47 != zero_zero_real ) ) )
% 0.20/0.47 = ( ! [Z3: real] :
% 0.20/0.47 ( ( ( ord_less_real @ Lb2 @ Z3 )
% 0.20/0.47 & ( ord_less_eq_real @ Z3 @ Ub ) )
% 0.20/0.47 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z3 ) ) )
% 0.20/0.47 | ! [Z3: real] :
% 0.20/0.47 ( ( ( ord_less_real @ Lb2 @ Z3 )
% 0.20/0.47 & ( ord_less_eq_real @ Z3 @ Ub ) )
% 0.20/0.47 => ( ord_less_real @ ( poly_real2 @ P @ Z3 ) @ zero_zero_real ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_eq_pos_or_neg_iff_1
% 0.20/0.47 thf(fact_79__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb2_O_A_092_060forall_062x_092_060le_062lb2_O_Asgn_A_Ipoly_Ap_Ax_J_A_061_Asgn__neg__inf_Ap_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.20/0.47 ~ ! [Lb22: real] :
% 0.20/0.47 ~ ! [X: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ X @ Lb22 )
% 0.20/0.47 => ( ( sgn_sgn_real @ ( poly_real2 @ p @ X ) )
% 0.20/0.47 = ( sturm_1076696862f_real @ p ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % \<open>\<And>thesis. (\<And>lb2. \<forall>x\<le>lb2. sgn (poly p x) = sgn_neg_inf p \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.20/0.47 thf(fact_80_le__zero__eq,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.20/0.47 = ( N = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_zero_eq
% 0.20/0.47 thf(fact_81_min_Obounded__iff,axiom,
% 0.20/0.47 ! [A: real,B: real,C: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) )
% 0.20/0.47 = ( ( ord_less_eq_real @ A @ B )
% 0.20/0.47 & ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.bounded_iff
% 0.20/0.47 thf(fact_82_min_Obounded__iff,axiom,
% 0.20/0.47 ! [A: nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
% 0.20/0.47 = ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 & ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.bounded_iff
% 0.20/0.47 thf(fact_83_reflect__poly__0,axiom,
% 0.20/0.47 ( ( reflect_poly_real @ zero_zero_poly_real )
% 0.20/0.47 = zero_zero_poly_real ) ).
% 0.20/0.47
% 0.20/0.47 % reflect_poly_0
% 0.20/0.47 thf(fact_84_reflect__poly__0,axiom,
% 0.20/0.47 ( ( reflect_poly_nat @ zero_zero_poly_nat )
% 0.20/0.47 = zero_zero_poly_nat ) ).
% 0.20/0.47
% 0.20/0.47 % reflect_poly_0
% 0.20/0.47 thf(fact_85_reflect__poly__0,axiom,
% 0.20/0.47 ( ( reflec1522834046y_real @ zero_z1423781445y_real )
% 0.20/0.47 = zero_z1423781445y_real ) ).
% 0.20/0.47
% 0.20/0.47 % reflect_poly_0
% 0.20/0.47 thf(fact_86_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: real,P3: real > $o] :
% 0.20/0.47 ( ( member_real @ A @ ( collect_real @ P3 ) )
% 0.20/0.47 = ( P3 @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_87_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A3: set_real] :
% 0.20/0.47 ( ( collect_real
% 0.20/0.47 @ ^ [X4: real] : ( member_real @ X4 @ A3 ) )
% 0.20/0.47 = A3 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_88_that,axiom,
% 0.20/0.47 ! [Lb2: real] :
% 0.20/0.47 ( ! [X3: real] :
% 0.20/0.47 ( ( ( poly_real2 @ p @ X3 )
% 0.20/0.47 = zero_zero_real )
% 0.20/0.47 => ( ord_less_real @ Lb2 @ X3 ) )
% 0.20/0.47 => ( ! [X3: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ X3 @ Lb2 )
% 0.20/0.47 => ( ( sgn_sgn_real @ ( poly_real2 @ p @ X3 ) )
% 0.20/0.47 = ( sturm_1076696862f_real @ p ) ) )
% 0.20/0.47 => thesis ) ) ).
% 0.20/0.47
% 0.20/0.47 % that
% 0.20/0.47 thf(fact_89_complete__real,axiom,
% 0.20/0.47 ! [S: set_real] :
% 0.20/0.47 ( ? [X: real] : ( member_real @ X @ S )
% 0.20/0.47 => ( ? [Z2: real] :
% 0.20/0.47 ! [X3: real] :
% 0.20/0.47 ( ( member_real @ X3 @ S )
% 0.20/0.47 => ( ord_less_eq_real @ X3 @ Z2 ) )
% 0.20/0.47 => ? [Y2: real] :
% 0.20/0.47 ( ! [X: real] :
% 0.20/0.47 ( ( member_real @ X @ S )
% 0.20/0.47 => ( ord_less_eq_real @ X @ Y2 ) )
% 0.20/0.47 & ! [Z2: real] :
% 0.20/0.47 ( ! [X3: real] :
% 0.20/0.47 ( ( member_real @ X3 @ S )
% 0.20/0.47 => ( ord_less_eq_real @ X3 @ Z2 ) )
% 0.20/0.47 => ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % complete_real
% 0.20/0.47 thf(fact_90_zero__le,axiom,
% 0.20/0.47 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 0.20/0.47
% 0.20/0.47 % zero_le
% 0.20/0.47 thf(fact_91_le__numeral__extra_I3_J,axiom,
% 0.20/0.47 ord_le1180086932y_real @ zero_zero_poly_real @ zero_zero_poly_real ).
% 0.20/0.47
% 0.20/0.47 % le_numeral_extra(3)
% 0.20/0.47 thf(fact_92_le__numeral__extra_I3_J,axiom,
% 0.20/0.47 ord_le893774876y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ).
% 0.20/0.47
% 0.20/0.47 % le_numeral_extra(3)
% 0.20/0.47 thf(fact_93_le__numeral__extra_I3_J,axiom,
% 0.20/0.47 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.20/0.47
% 0.20/0.47 % le_numeral_extra(3)
% 0.20/0.47 thf(fact_94_le__numeral__extra_I3_J,axiom,
% 0.20/0.47 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.20/0.47
% 0.20/0.47 % le_numeral_extra(3)
% 0.20/0.47 thf(fact_95_less__eq__real__def,axiom,
% 0.20/0.47 ( ord_less_eq_real
% 0.20/0.47 = ( ^ [X4: real,Y3: real] :
% 0.20/0.47 ( ( ord_less_real @ X4 @ Y3 )
% 0.20/0.47 | ( X4 = Y3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_eq_real_def
% 0.20/0.47 thf(fact_96_min_Omono,axiom,
% 0.20/0.47 ! [A: real,C: real,B: real,D: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ A @ C )
% 0.20/0.47 => ( ( ord_less_eq_real @ B @ D )
% 0.20/0.47 => ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ ( ord_min_real @ C @ D ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.mono
% 0.20/0.47 thf(fact_97_min_Omono,axiom,
% 0.20/0.47 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ C )
% 0.20/0.47 => ( ( ord_less_eq_nat @ B @ D )
% 0.20/0.47 => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.mono
% 0.20/0.47 thf(fact_98_min_OorderE,axiom,
% 0.20/0.47 ! [A: real,B: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.20/0.47 => ( A
% 0.20/0.47 = ( ord_min_real @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.orderE
% 0.20/0.47 thf(fact_99_min_OorderE,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 => ( A
% 0.20/0.47 = ( ord_min_nat @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.orderE
% 0.20/0.47 thf(fact_100_min_OorderI,axiom,
% 0.20/0.47 ! [A: real,B: real] :
% 0.20/0.47 ( ( A
% 0.20/0.47 = ( ord_min_real @ A @ B ) )
% 0.20/0.47 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.orderI
% 0.20/0.47 thf(fact_101_min_OorderI,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( A
% 0.20/0.47 = ( ord_min_nat @ A @ B ) )
% 0.20/0.47 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.orderI
% 0.20/0.47 thf(fact_102_min_Oabsorb1,axiom,
% 0.20/0.47 ! [A: real,B: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.20/0.47 => ( ( ord_min_real @ A @ B )
% 0.20/0.47 = A ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.absorb1
% 0.20/0.47 thf(fact_103_min_Oabsorb1,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 => ( ( ord_min_nat @ A @ B )
% 0.20/0.47 = A ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.absorb1
% 0.20/0.47 thf(fact_104_min_Oabsorb2,axiom,
% 0.20/0.47 ! [B: real,A: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ B @ A )
% 0.20/0.47 => ( ( ord_min_real @ A @ B )
% 0.20/0.47 = B ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.absorb2
% 0.20/0.47 thf(fact_105_min_Oabsorb2,axiom,
% 0.20/0.47 ! [B: nat,A: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.47 => ( ( ord_min_nat @ A @ B )
% 0.20/0.47 = B ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.absorb2
% 0.20/0.47 thf(fact_106_min_OboundedE,axiom,
% 0.20/0.47 ! [A: real,B: real,C: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) )
% 0.20/0.47 => ~ ( ( ord_less_eq_real @ A @ B )
% 0.20/0.47 => ~ ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.boundedE
% 0.20/0.47 thf(fact_107_min_OboundedE,axiom,
% 0.20/0.47 ! [A: nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
% 0.20/0.47 => ~ ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.boundedE
% 0.20/0.47 thf(fact_108_min_OboundedI,axiom,
% 0.20/0.47 ! [A: real,B: real,C: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.20/0.47 => ( ( ord_less_eq_real @ A @ C )
% 0.20/0.47 => ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.boundedI
% 0.20/0.47 thf(fact_109_min_OboundedI,axiom,
% 0.20/0.47 ! [A: nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 => ( ( ord_less_eq_nat @ A @ C )
% 0.20/0.47 => ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.boundedI
% 0.20/0.47 thf(fact_110_min_Oorder__iff,axiom,
% 0.20/0.47 ( ord_less_eq_real
% 0.20/0.47 = ( ^ [A2: real,B2: real] :
% 0.20/0.47 ( A2
% 0.20/0.47 = ( ord_min_real @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.order_iff
% 0.20/0.47 thf(fact_111_min_Oorder__iff,axiom,
% 0.20/0.47 ( ord_less_eq_nat
% 0.20/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.20/0.47 ( A2
% 0.20/0.47 = ( ord_min_nat @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.order_iff
% 0.20/0.47 thf(fact_112_min_Ocobounded1,axiom,
% 0.20/0.47 ! [A: real,B: real] : ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ A ) ).
% 0.20/0.47
% 0.20/0.47 % min.cobounded1
% 0.20/0.47 thf(fact_113_min_Ocobounded1,axiom,
% 0.20/0.47 ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% 0.20/0.47
% 0.20/0.47 % min.cobounded1
% 0.20/0.47 thf(fact_114_min_Ocobounded2,axiom,
% 0.20/0.47 ! [A: real,B: real] : ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ B ) ).
% 0.20/0.47
% 0.20/0.47 % min.cobounded2
% 0.20/0.47 thf(fact_115_min_Ocobounded2,axiom,
% 0.20/0.47 ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% 0.20/0.47
% 0.20/0.47 % min.cobounded2
% 0.20/0.47 thf(fact_116_min_Oabsorb__iff1,axiom,
% 0.20/0.47 ( ord_less_eq_real
% 0.20/0.47 = ( ^ [A2: real,B2: real] :
% 0.20/0.47 ( ( ord_min_real @ A2 @ B2 )
% 0.20/0.47 = A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.absorb_iff1
% 0.20/0.47 thf(fact_117_min_Oabsorb__iff1,axiom,
% 0.20/0.47 ( ord_less_eq_nat
% 0.20/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.20/0.47 ( ( ord_min_nat @ A2 @ B2 )
% 0.20/0.47 = A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.absorb_iff1
% 0.20/0.47 thf(fact_118_min_Oabsorb__iff2,axiom,
% 0.20/0.47 ( ord_less_eq_real
% 0.20/0.47 = ( ^ [B2: real,A2: real] :
% 0.20/0.47 ( ( ord_min_real @ A2 @ B2 )
% 0.20/0.47 = B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.absorb_iff2
% 0.20/0.47 thf(fact_119_min_Oabsorb__iff2,axiom,
% 0.20/0.47 ( ord_less_eq_nat
% 0.20/0.47 = ( ^ [B2: nat,A2: nat] :
% 0.20/0.47 ( ( ord_min_nat @ A2 @ B2 )
% 0.20/0.47 = B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.absorb_iff2
% 0.20/0.47 thf(fact_120_min_OcoboundedI1,axiom,
% 0.20/0.47 ! [A: real,C: real,B: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ A @ C )
% 0.20/0.47 => ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.coboundedI1
% 0.20/0.47 thf(fact_121_min_OcoboundedI1,axiom,
% 0.20/0.47 ! [A: nat,C: nat,B: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ C )
% 0.20/0.47 => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.coboundedI1
% 0.20/0.47 thf(fact_122_min_OcoboundedI2,axiom,
% 0.20/0.47 ! [B: real,C: real,A: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ B @ C )
% 0.20/0.47 => ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.coboundedI2
% 0.20/0.47 thf(fact_123_min_OcoboundedI2,axiom,
% 0.20/0.47 ! [B: nat,C: nat,A: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.47 => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 % min.coboundedI2
% 0.20/0.47 thf(fact_124_min__le__iff__disj,axiom,
% 0.20/0.47 ! [X2: real,Y: real,Z: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ ( ord_min_real @ X2 @ Y ) @ Z )
% 0.20/0.47 = ( ( ord_less_eq_real @ X2 @ Z )
% 0.20/0.47 | ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_le_iff_disj
% 0.20/0.47 thf(fact_125_min__le__iff__disj,axiom,
% 0.20/0.47 ! [X2: nat,Y: nat,Z: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ ( ord_min_nat @ X2 @ Y ) @ Z )
% 0.20/0.47 = ( ( ord_less_eq_nat @ X2 @ Z )
% 0.20/0.47 | ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_le_iff_disj
% 0.20/0.47 thf(fact_126_order__0I,axiom,
% 0.20/0.47 ! [P: poly_real,A: real] :
% 0.20/0.47 ( ( ( poly_real2 @ P @ A )
% 0.20/0.47 != zero_zero_real )
% 0.20/0.47 => ( ( order_real @ A @ P )
% 0.20/0.47 = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_0I
% 0.20/0.47 thf(fact_127_order__0I,axiom,
% 0.20/0.47 ! [P: poly_poly_real,A: poly_real] :
% 0.20/0.47 ( ( ( poly_poly_real2 @ P @ A )
% 0.20/0.47 != zero_zero_poly_real )
% 0.20/0.47 => ( ( order_poly_real @ A @ P )
% 0.20/0.47 = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_0I
% 0.20/0.47 thf(fact_128_order__0I,axiom,
% 0.20/0.47 ! [P: poly_poly_poly_real,A: poly_poly_real] :
% 0.20/0.47 ( ( ( poly_poly_poly_real2 @ P @ A )
% 0.20/0.47 != zero_z1423781445y_real )
% 0.20/0.47 => ( ( order_poly_poly_real @ A @ P )
% 0.20/0.47 = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_0I
% 0.20/0.47 thf(fact_129_not__eq__pos__or__neg__iff__2,axiom,
% 0.20/0.47 ! [Lb2: real,Ub: real,P: poly_real] :
% 0.20/0.47 ( ( ! [Z3: real] :
% 0.20/0.47 ( ( ( ord_less_eq_real @ Lb2 @ Z3 )
% 0.20/0.47 & ( ord_less_real @ Z3 @ Ub ) )
% 0.20/0.47 => ( ( poly_real2 @ P @ Z3 )
% 0.20/0.47 != zero_zero_real ) ) )
% 0.20/0.47 = ( ! [Z3: real] :
% 0.20/0.47 ( ( ( ord_less_eq_real @ Lb2 @ Z3 )
% 0.20/0.47 & ( ord_less_real @ Z3 @ Ub ) )
% 0.20/0.47 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z3 ) ) )
% 0.20/0.47 | ! [Z3: real] :
% 0.20/0.47 ( ( ( ord_less_eq_real @ Lb2 @ Z3 )
% 0.20/0.47 & ( ord_less_real @ Z3 @ Ub ) )
% 0.20/0.47 => ( ord_less_real @ ( poly_real2 @ P @ Z3 ) @ zero_zero_real ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_eq_pos_or_neg_iff_2
% 0.20/0.47 thf(fact_130_zero__le__sgn__iff,axiom,
% 0.20/0.47 ! [X2: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X2 ) )
% 0.20/0.47 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_le_sgn_iff
% 0.20/0.47 thf(fact_131_sgn__le__0__iff,axiom,
% 0.20/0.47 ! [X2: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X2 ) @ zero_zero_real )
% 0.20/0.47 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_le_0_iff
% 0.20/0.47 thf(fact_132_sgn__greater,axiom,
% 0.20/0.47 ! [A: poly_real] :
% 0.20/0.47 ( ( ord_less_poly_real @ zero_zero_poly_real @ ( sgn_sgn_poly_real @ A ) )
% 0.20/0.47 = ( ord_less_poly_real @ zero_zero_poly_real @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_greater
% 0.20/0.47 thf(fact_133_sgn__greater,axiom,
% 0.20/0.47 ! [A: poly_poly_real] :
% 0.20/0.47 ( ( ord_le38482960y_real @ zero_z1423781445y_real @ ( sgn_sg2128174761y_real @ A ) )
% 0.20/0.47 = ( ord_le38482960y_real @ zero_z1423781445y_real @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_greater
% 0.20/0.47 thf(fact_134_sgn__greater,axiom,
% 0.20/0.47 ! [A: real] :
% 0.20/0.47 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 0.20/0.47 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_greater
% 0.20/0.47 thf(fact_135_sgn__less,axiom,
% 0.20/0.47 ! [A: poly_real] :
% 0.20/0.47 ( ( ord_less_poly_real @ ( sgn_sgn_poly_real @ A ) @ zero_zero_poly_real )
% 0.20/0.47 = ( ord_less_poly_real @ A @ zero_zero_poly_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_less
% 0.20/0.47 thf(fact_136_sgn__less,axiom,
% 0.20/0.47 ! [A: poly_poly_real] :
% 0.20/0.47 ( ( ord_le38482960y_real @ ( sgn_sg2128174761y_real @ A ) @ zero_z1423781445y_real )
% 0.20/0.47 = ( ord_le38482960y_real @ A @ zero_z1423781445y_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_less
% 0.20/0.47 thf(fact_137_sgn__less,axiom,
% 0.20/0.47 ! [A: real] :
% 0.20/0.47 ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 0.20/0.47 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_less
% 0.20/0.47 thf(fact_138_root__ub,axiom,
% 0.20/0.47 ! [P: poly_real] :
% 0.20/0.47 ( ( P != zero_zero_poly_real )
% 0.20/0.47 => ~ ! [Ub2: real] :
% 0.20/0.47 ( ! [X: real] :
% 0.20/0.47 ( ( ( poly_real2 @ P @ X )
% 0.20/0.47 = zero_zero_real )
% 0.20/0.47 => ( ord_less_real @ X @ Ub2 ) )
% 0.20/0.47 => ~ ! [X: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ Ub2 @ X )
% 0.20/0.47 => ( ( sgn_sgn_real @ ( poly_real2 @ P @ X ) )
% 0.20/0.47 = ( sturm_1308388506f_real @ P ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % root_ub
% 0.20/0.47 thf(fact_139_sgn__0,axiom,
% 0.20/0.47 ( ( sgn_sgn_real @ zero_zero_real )
% 0.20/0.47 = zero_zero_real ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_0
% 0.20/0.47 thf(fact_140_sgn__0,axiom,
% 0.20/0.47 ( ( sgn_sgn_poly_real @ zero_zero_poly_real )
% 0.20/0.47 = zero_zero_poly_real ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_0
% 0.20/0.47 thf(fact_141_sgn__0,axiom,
% 0.20/0.47 ( ( sgn_sg2128174761y_real @ zero_z1423781445y_real )
% 0.20/0.47 = zero_z1423781445y_real ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_0
% 0.20/0.47 thf(fact_142_sgn__zero,axiom,
% 0.20/0.47 ( ( sgn_sgn_real @ zero_zero_real )
% 0.20/0.47 = zero_zero_real ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_zero
% 0.20/0.47 thf(fact_143_sgn__sgn,axiom,
% 0.20/0.47 ! [A: real] :
% 0.20/0.47 ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
% 0.20/0.47 = ( sgn_sgn_real @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_sgn
% 0.20/0.47 thf(fact_144_linorder__neqE__linordered__idom,axiom,
% 0.20/0.47 ! [X2: real,Y: real] :
% 0.20/0.47 ( ( X2 != Y )
% 0.20/0.47 => ( ~ ( ord_less_real @ X2 @ Y )
% 0.20/0.47 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % linorder_neqE_linordered_idom
% 0.20/0.47 thf(fact_145_sgn__eq__0__iff,axiom,
% 0.20/0.47 ! [A: real] :
% 0.20/0.47 ( ( ( sgn_sgn_real @ A )
% 0.20/0.47 = zero_zero_real )
% 0.20/0.47 = ( A = zero_zero_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_eq_0_iff
% 0.20/0.47 thf(fact_146_sgn__eq__0__iff,axiom,
% 0.20/0.47 ! [A: poly_real] :
% 0.20/0.47 ( ( ( sgn_sgn_poly_real @ A )
% 0.20/0.47 = zero_zero_poly_real )
% 0.20/0.47 = ( A = zero_zero_poly_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_eq_0_iff
% 0.20/0.47 thf(fact_147_sgn__eq__0__iff,axiom,
% 0.20/0.47 ! [A: poly_poly_real] :
% 0.20/0.47 ( ( ( sgn_sg2128174761y_real @ A )
% 0.20/0.47 = zero_z1423781445y_real )
% 0.20/0.47 = ( A = zero_z1423781445y_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_eq_0_iff
% 0.20/0.47 thf(fact_148_sgn__0__0,axiom,
% 0.20/0.47 ! [A: real] :
% 0.20/0.47 ( ( ( sgn_sgn_real @ A )
% 0.20/0.47 = zero_zero_real )
% 0.20/0.47 = ( A = zero_zero_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_0_0
% 0.20/0.47 thf(fact_149_sgn__0__0,axiom,
% 0.20/0.47 ! [A: poly_real] :
% 0.20/0.47 ( ( ( sgn_sgn_poly_real @ A )
% 0.20/0.47 = zero_zero_poly_real )
% 0.20/0.47 = ( A = zero_zero_poly_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_0_0
% 0.20/0.47 thf(fact_150_sgn__0__0,axiom,
% 0.20/0.47 ! [A: poly_poly_real] :
% 0.20/0.47 ( ( ( sgn_sg2128174761y_real @ A )
% 0.20/0.47 = zero_z1423781445y_real )
% 0.20/0.47 = ( A = zero_z1423781445y_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_0_0
% 0.20/0.47 thf(fact_151_sgn__zero__iff,axiom,
% 0.20/0.47 ! [X2: real] :
% 0.20/0.47 ( ( ( sgn_sgn_real @ X2 )
% 0.20/0.47 = zero_zero_real )
% 0.20/0.47 = ( X2 = zero_zero_real ) ) ).
% 0.20/0.47
% 0.20/0.47 % sgn_zero_iff
% 0.20/0.47 thf(fact_152_le0,axiom,
% 0.20/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.20/0.47
% 0.20/0.47 % le0
% 0.20/0.47 thf(fact_153_bot__nat__0_Oextremum,axiom,
% 0.20/0.47 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.20/0.47
% 0.20/0.47 % bot_nat_0.extremum
% 0.20/0.47 thf(fact_154_neq0__conv,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( N != zero_zero_nat )
% 0.20/0.47 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % neq0_conv
% 0.20/0.47 thf(fact_155_less__nat__zero__code,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.20/0.47
% 0.20/0.47 % less_nat_zero_code
% 0.20/0.47 thf(fact_156_bot__nat__0_Onot__eq__extremum,axiom,
% 0.20/0.47 ! [A: nat] :
% 0.20/0.47 ( ( A != zero_zero_nat )
% 0.20/0.47 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_nat_0.not_eq_extremum
% 0.20/0.47 thf(fact_157_min__0R,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.20/0.47 = zero_zero_nat ) ).
% 0.20/0.47
% 0.20/0.47 % min_0R
% 0.20/0.47 thf(fact_158_min__0L,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.20/0.47 = zero_zero_nat ) ).
% 0.20/0.47
% 0.20/0.47 % min_0L
% 0.20/0.47 thf(fact_159_le__refl,axiom,
% 0.20/0.47 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.20/0.47
% 0.20/0.47 % le_refl
% 0.20/0.47 thf(fact_160_le__trans,axiom,
% 0.20/0.47 ! [I: nat,J: nat,K: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.47 => ( ( ord_less_eq_nat @ J @ K )
% 0.20/0.47 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_trans
% 0.20/0.47 thf(fact_161_eq__imp__le,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( M = N )
% 0.20/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_imp_le
% 0.20/0.47 thf(fact_162_le__antisym,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.47 => ( ( ord_less_eq_nat @ N @ M )
% 0.20/0.47 => ( M = N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_antisym
% 0.20/0.47 thf(fact_163_nat__le__linear,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.47 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_le_linear
% 0.20/0.47 thf(fact_164_Nat_Oex__has__greatest__nat,axiom,
% 0.20/0.47 ! [P3: nat > $o,K: nat,B: nat] :
% 0.20/0.47 ( ( P3 @ K )
% 0.20/0.47 => ( ! [Y2: nat] :
% 0.20/0.47 ( ( P3 @ Y2 )
% 0.20/0.47 => ( ord_less_eq_nat @ Y2 @ B ) )
% 0.20/0.47 => ? [X3: nat] :
% 0.20/0.47 ( ( P3 @ X3 )
% 0.20/0.47 & ! [Y4: nat] :
% 0.20/0.47 ( ( P3 @ Y4 )
% 0.20/0.47 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Nat.ex_has_greatest_nat
% 0.20/0.47 thf(fact_165_less__mono__imp__le__mono,axiom,
% 0.20/0.47 ! [F: nat > nat,I: nat,J: nat] :
% 0.20/0.47 ( ! [I2: nat,J2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I2 @ J2 )
% 0.20/0.47 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.20/0.47 => ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.47 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_mono_imp_le_mono
% 0.20/0.47 thf(fact_166_le__neq__implies__less,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.47 => ( ( M != N )
% 0.20/0.47 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_neq_implies_less
% 0.20/0.47 thf(fact_167_less__or__eq__imp__le,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ( ord_less_nat @ M @ N )
% 0.20/0.47 | ( M = N ) )
% 0.20/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_or_eq_imp_le
% 0.20/0.47 thf(fact_168_le__eq__less__or__eq,axiom,
% 0.20/0.47 ( ord_less_eq_nat
% 0.20/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.20/0.47 | ( M2 = N2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_eq_less_or_eq
% 0.20/0.47 thf(fact_169_less__imp__le__nat,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less_nat @ M @ N )
% 0.20/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_imp_le_nat
% 0.20/0.47 thf(fact_170_nat__less__le,axiom,
% 0.20/0.47 ( ord_less_nat
% 0.20/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.20/0.47 & ( M2 != N2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_less_le
% 0.20/0.47 thf(fact_171_linorder__neqE__nat,axiom,
% 0.20/0.47 ! [X2: nat,Y: nat] :
% 0.20/0.47 ( ( X2 != Y )
% 0.20/0.47 => ( ~ ( ord_less_nat @ X2 @ Y )
% 0.20/0.47 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % linorder_neqE_nat
% 0.20/0.47 thf(fact_172_infinite__descent,axiom,
% 0.20/0.47 ! [P3: nat > $o,N: nat] :
% 0.20/0.47 ( ! [N3: nat] :
% 0.20/0.47 ( ~ ( P3 @ N3 )
% 0.20/0.47 => ? [M3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ M3 @ N3 )
% 0.20/0.47 & ~ ( P3 @ M3 ) ) )
% 0.20/0.47 => ( P3 @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % infinite_descent
% 0.20/0.47 thf(fact_173_nat__less__induct,axiom,
% 0.20/0.47 ! [P3: nat > $o,N: nat] :
% 0.20/0.47 ( ! [N3: nat] :
% 0.20/0.47 ( ! [M3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ M3 @ N3 )
% 0.20/0.47 => ( P3 @ M3 ) )
% 0.20/0.47 => ( P3 @ N3 ) )
% 0.20/0.47 => ( P3 @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_less_induct
% 0.20/0.47 thf(fact_174_less__irrefl__nat,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.20/0.47
% 0.20/0.47 % less_irrefl_nat
% 0.20/0.47 thf(fact_175_less__not__refl3,axiom,
% 0.20/0.47 ! [S2: nat,T: nat] :
% 0.20/0.47 ( ( ord_less_nat @ S2 @ T )
% 0.20/0.47 => ( S2 != T ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_not_refl3
% 0.20/0.47 thf(fact_176_less__not__refl2,axiom,
% 0.20/0.47 ! [N: nat,M: nat] :
% 0.20/0.47 ( ( ord_less_nat @ N @ M )
% 0.20/0.47 => ( M != N ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_not_refl2
% 0.20/0.47 thf(fact_177_less__not__refl,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.20/0.47
% 0.20/0.47 % less_not_refl
% 0.20/0.47 thf(fact_178_nat__neq__iff,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( M != N )
% 0.20/0.47 = ( ( ord_less_nat @ M @ N )
% 0.20/0.47 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_neq_iff
% 0.20/0.47 thf(fact_179_bot__nat__0_Oextremum__strict,axiom,
% 0.20/0.47 ! [A: nat] :
% 0.20/0.47 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.20/0.47
% 0.20/0.47 % bot_nat_0.extremum_strict
% 0.20/0.47 thf(fact_180_infinite__descent0,axiom,
% 0.20/0.47 ! [P3: nat > $o,N: nat] :
% 0.20/0.47 ( ( P3 @ zero_zero_nat )
% 0.20/0.47 => ( ! [N3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.20/0.47 => ( ~ ( P3 @ N3 )
% 0.20/0.47 => ? [M3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ M3 @ N3 )
% 0.20/0.47 & ~ ( P3 @ M3 ) ) ) )
% 0.20/0.47 => ( P3 @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % infinite_descent0
% 0.20/0.47 thf(fact_181_gr__implies__not0,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less_nat @ M @ N )
% 0.20/0.47 => ( N != zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % gr_implies_not0
% 0.20/0.47 thf(fact_182_less__zeroE,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.20/0.47
% 0.20/0.47 % less_zeroE
% 0.20/0.47 thf(fact_183_not__less0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.20/0.47
% 0.20/0.47 % not_less0
% 0.20/0.47 thf(fact_184_not__gr0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.20/0.47 = ( N = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_gr0
% 0.20/0.47 thf(fact_185_gr0I,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( N != zero_zero_nat )
% 0.20/0.47 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % gr0I
% 0.20/0.47 thf(fact_186_less__eq__nat_Osimps_I1_J,axiom,
% 0.20/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.20/0.47
% 0.20/0.47 % less_eq_nat.simps(1)
% 0.20/0.47 thf(fact_187_le__0__eq,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.20/0.47 = ( N = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_0_eq
% 0.20/0.47 thf(fact_188_ex__least__nat__le,axiom,
% 0.20/0.47 ! [P3: nat > $o,N: nat] :
% 0.20/0.47 ( ( P3 @ N )
% 0.20/0.47 => ( ~ ( P3 @ zero_zero_nat )
% 0.20/0.47 => ? [K2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ K2 @ N )
% 0.20/0.47 & ! [I3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I3 @ K2 )
% 0.20/0.47 => ~ ( P3 @ I3 ) )
% 0.20/0.47 & ( P3 @ K2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ex_least_nat_le
% 0.20/0.47 thf(fact_189_bot__nat__0_Oextremum__unique,axiom,
% 0.20/0.47 ! [A: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.20/0.47 = ( A = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_nat_0.extremum_unique
% 0.20/0.47 thf(fact_190_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.20/0.47 ! [A: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.20/0.47 => ( A = zero_zero_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_nat_0.extremum_uniqueI
% 0.20/0.47 thf(fact_191_order__refl,axiom,
% 0.20/0.47 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 0.20/0.47
% 0.20/0.47 % order_refl
% 0.20/0.47 thf(fact_192_order__refl,axiom,
% 0.20/0.47 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.20/0.47
% 0.20/0.47 % order_refl
% 0.20/0.47 thf(fact_193_min__absorb1,axiom,
% 0.20/0.47 ! [X2: real,Y: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ X2 @ Y )
% 0.20/0.47 => ( ( ord_min_real @ X2 @ Y )
% 0.20/0.47 = X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_absorb1
% 0.20/0.47 thf(fact_194_min__absorb1,axiom,
% 0.20/0.47 ! [X2: nat,Y: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.20/0.47 => ( ( ord_min_nat @ X2 @ Y )
% 0.20/0.47 = X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_absorb1
% 0.20/0.47 thf(fact_195_min__absorb2,axiom,
% 0.20/0.47 ! [Y: real,X2: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ Y @ X2 )
% 0.20/0.47 => ( ( ord_min_real @ X2 @ Y )
% 0.20/0.47 = Y ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_absorb2
% 0.20/0.47 thf(fact_196_min__absorb2,axiom,
% 0.20/0.47 ! [Y: nat,X2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ Y @ X2 )
% 0.20/0.47 => ( ( ord_min_nat @ X2 @ Y )
% 0.20/0.47 = Y ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_absorb2
% 0.20/0.47 thf(fact_197_min__def,axiom,
% 0.20/0.47 ( ord_min_real
% 0.20/0.47 = ( ^ [A2: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_def
% 0.20/0.47 thf(fact_198_min__def,axiom,
% 0.20/0.47 ( ord_min_nat
% 0.20/0.47 = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % min_def
% 0.20/0.47 thf(fact_199_nat__descend__induct,axiom,
% 0.20/0.47 ! [N: nat,P3: nat > $o,M: nat] :
% 0.20/0.47 ( ! [K2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ N @ K2 )
% 0.20/0.47 => ( P3 @ K2 ) )
% 0.20/0.47 => ( ! [K2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ K2 @ N )
% 0.20/0.47 => ( ! [I3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ K2 @ I3 )
% 0.20/0.47 => ( P3 @ I3 ) )
% 0.20/0.47 => ( P3 @ K2 ) ) )
% 0.20/0.47 => ( P3 @ M ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_descend_induct
% 0.20/0.47 thf(fact_200_dual__order_Oantisym,axiom,
% 0.20/0.47 ! [B: real,A: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ B @ A )
% 0.20/0.47 => ( ( ord_less_eq_real @ A @ B )
% 0.20/0.47 => ( A = B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.antisym
% 0.20/0.47 thf(fact_201_dual__order_Oantisym,axiom,
% 0.20/0.47 ! [B: nat,A: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.47 => ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 => ( A = B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.antisym
% 0.20/0.47 thf(fact_202_dual__order_Oeq__iff,axiom,
% 0.20/0.47 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.20/0.47 = ( ^ [A2: real,B2: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ B2 @ A2 )
% 0.20/0.47 & ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.eq_iff
% 0.20/0.47 thf(fact_203_dual__order_Oeq__iff,axiom,
% 0.20/0.47 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.20/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.20/0.47 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.eq_iff
% 0.20/0.47 thf(fact_204_dual__order_Otrans,axiom,
% 0.20/0.47 ! [B: real,A: real,C: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ B @ A )
% 0.20/0.47 => ( ( ord_less_eq_real @ C @ B )
% 0.20/0.47 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.trans
% 0.20/0.47 thf(fact_205_dual__order_Otrans,axiom,
% 0.20/0.47 ! [B: nat,A: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.47 => ( ( ord_less_eq_nat @ C @ B )
% 0.20/0.47 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.trans
% 0.20/0.47 thf(fact_206_linorder__wlog,axiom,
% 0.20/0.47 ! [P3: real > real > $o,A: real,B: real] :
% 0.20/0.47 ( ! [A4: real,B3: real] :
% 0.20/0.47 ( ( ord_less_eq_real @ A4 @ B3 )
% 0.31/0.47 => ( P3 @ A4 @ B3 ) )
% 0.31/0.47 => ( ! [A4: real,B3: real] :
% 0.31/0.47 ( ( P3 @ B3 @ A4 )
% 0.31/0.47 => ( P3 @ A4 @ B3 ) )
% 0.31/0.47 => ( P3 @ A @ B ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_wlog
% 0.31/0.47 thf(fact_207_linorder__wlog,axiom,
% 0.31/0.47 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 0.31/0.47 ( ! [A4: nat,B3: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A4 @ B3 )
% 0.31/0.47 => ( P3 @ A4 @ B3 ) )
% 0.31/0.47 => ( ! [A4: nat,B3: nat] :
% 0.31/0.47 ( ( P3 @ B3 @ A4 )
% 0.31/0.47 => ( P3 @ A4 @ B3 ) )
% 0.31/0.47 => ( P3 @ A @ B ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_wlog
% 0.31/0.47 thf(fact_208_dual__order_Orefl,axiom,
% 0.31/0.47 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.refl
% 0.31/0.47 thf(fact_209_dual__order_Orefl,axiom,
% 0.31/0.47 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.refl
% 0.31/0.47 thf(fact_210_order__trans,axiom,
% 0.31/0.47 ! [X2: real,Y: real,Z: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X2 @ Y )
% 0.31/0.47 => ( ( ord_less_eq_real @ Y @ Z )
% 0.31/0.47 => ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_trans
% 0.31/0.47 thf(fact_211_order__trans,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat,Z: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.31/0.47 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.31/0.47 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_trans
% 0.31/0.47 thf(fact_212_order__class_Oorder_Oantisym,axiom,
% 0.31/0.47 ! [A: real,B: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.31/0.47 => ( ( ord_less_eq_real @ B @ A )
% 0.31/0.47 => ( A = B ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_class.order.antisym
% 0.31/0.47 thf(fact_213_order__class_Oorder_Oantisym,axiom,
% 0.31/0.47 ! [A: nat,B: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.47 => ( ( ord_less_eq_nat @ B @ A )
% 0.31/0.47 => ( A = B ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_class.order.antisym
% 0.31/0.47 thf(fact_214_ord__le__eq__trans,axiom,
% 0.31/0.47 ! [A: real,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.31/0.47 => ( ( B = C )
% 0.31/0.47 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_trans
% 0.31/0.47 thf(fact_215_ord__le__eq__trans,axiom,
% 0.31/0.47 ! [A: nat,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.47 => ( ( B = C )
% 0.31/0.47 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_trans
% 0.31/0.47 thf(fact_216_ord__eq__le__trans,axiom,
% 0.31/0.47 ! [A: real,B: real,C: real] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.31/0.47 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_trans
% 0.31/0.47 thf(fact_217_ord__eq__le__trans,axiom,
% 0.31/0.47 ! [A: nat,B: nat,C: nat] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.47 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_trans
% 0.31/0.47 thf(fact_218_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.47 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.31/0.47 = ( ^ [A2: real,B2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ A2 @ B2 )
% 0.31/0.47 & ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_class.order.eq_iff
% 0.31/0.47 thf(fact_219_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.47 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.31/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.31/0.47 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_class.order.eq_iff
% 0.31/0.47 thf(fact_220_antisym__conv,axiom,
% 0.31/0.47 ! [Y: real,X2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ Y @ X2 )
% 0.31/0.47 => ( ( ord_less_eq_real @ X2 @ Y )
% 0.31/0.47 = ( X2 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv
% 0.31/0.47 thf(fact_221_antisym__conv,axiom,
% 0.31/0.47 ! [Y: nat,X2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ Y @ X2 )
% 0.31/0.47 => ( ( ord_less_eq_nat @ X2 @ Y )
% 0.31/0.47 = ( X2 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv
% 0.31/0.47 thf(fact_222_le__cases3,axiom,
% 0.31/0.47 ! [X2: real,Y: real,Z: real] :
% 0.31/0.47 ( ( ( ord_less_eq_real @ X2 @ Y )
% 0.31/0.47 => ~ ( ord_less_eq_real @ Y @ Z ) )
% 0.31/0.47 => ( ( ( ord_less_eq_real @ Y @ X2 )
% 0.31/0.47 => ~ ( ord_less_eq_real @ X2 @ Z ) )
% 0.31/0.47 => ( ( ( ord_less_eq_real @ X2 @ Z )
% 0.31/0.47 => ~ ( ord_less_eq_real @ Z @ Y ) )
% 0.31/0.47 => ( ( ( ord_less_eq_real @ Z @ Y )
% 0.31/0.47 => ~ ( ord_less_eq_real @ Y @ X2 ) )
% 0.31/0.47 => ( ( ( ord_less_eq_real @ Y @ Z )
% 0.31/0.47 => ~ ( ord_less_eq_real @ Z @ X2 ) )
% 0.31/0.47 => ~ ( ( ord_less_eq_real @ Z @ X2 )
% 0.31/0.47 => ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_cases3
% 0.31/0.47 thf(fact_223_le__cases3,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat,Z: nat] :
% 0.31/0.47 ( ( ( ord_less_eq_nat @ X2 @ Y )
% 0.31/0.47 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 0.31/0.47 => ( ( ( ord_less_eq_nat @ Y @ X2 )
% 0.31/0.47 => ~ ( ord_less_eq_nat @ X2 @ Z ) )
% 0.31/0.47 => ( ( ( ord_less_eq_nat @ X2 @ Z )
% 0.31/0.47 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 0.31/0.47 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 0.31/0.47 => ~ ( ord_less_eq_nat @ Y @ X2 ) )
% 0.31/0.47 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 0.31/0.47 => ~ ( ord_less_eq_nat @ Z @ X2 ) )
% 0.31/0.47 => ~ ( ( ord_less_eq_nat @ Z @ X2 )
% 0.31/0.47 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_cases3
% 0.31/0.47 thf(fact_224_order_Otrans,axiom,
% 0.31/0.47 ! [A: real,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.31/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.31/0.47 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.trans
% 0.31/0.47 thf(fact_225_order_Otrans,axiom,
% 0.31/0.47 ! [A: nat,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.47 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.trans
% 0.31/0.47 thf(fact_226_le__cases,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ~ ( ord_less_eq_real @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_cases
% 0.31/0.47 thf(fact_227_le__cases,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ~ ( ord_less_eq_nat @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_cases
% 0.31/0.47 thf(fact_228_eq__refl,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( X2 = Y )
% 0.31/0.47 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_refl
% 0.31/0.47 thf(fact_229_eq__refl,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( X2 = Y )
% 0.31/0.47 => ( ord_less_eq_nat @ X2 @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_refl
% 0.31/0.47 thf(fact_230_linear,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X2 @ Y )
% 0.31/0.47 | ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % linear
% 0.31/0.47 thf(fact_231_linear,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.31/0.47 | ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % linear
% 0.31/0.47 thf(fact_232_antisym,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X2 @ Y )
% 0.31/0.47 => ( ( ord_less_eq_real @ Y @ X2 )
% 0.31/0.47 => ( X2 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym
% 0.31/0.47 thf(fact_233_antisym,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.31/0.47 => ( ( ord_less_eq_nat @ Y @ X2 )
% 0.31/0.47 => ( X2 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym
% 0.31/0.47 thf(fact_234_eq__iff,axiom,
% 0.31/0.47 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.31/0.47 = ( ^ [X4: real,Y3: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.31/0.47 & ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_iff
% 0.31/0.47 thf(fact_235_eq__iff,axiom,
% 0.31/0.47 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.31/0.47 = ( ^ [X4: nat,Y3: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.31/0.47 & ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_iff
% 0.31/0.47 thf(fact_236_ord__le__eq__subst,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.31/0.47 => ( ( ( F @ B )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_subst
% 0.31/0.47 thf(fact_237_ord__le__eq__subst,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.31/0.47 => ( ( ( F @ B )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_subst
% 0.31/0.47 thf(fact_238_ord__le__eq__subst,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.47 => ( ( ( F @ B )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_subst
% 0.31/0.47 thf(fact_239_ord__le__eq__subst,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.47 => ( ( ( F @ B )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_subst
% 0.31/0.47 thf(fact_240_ord__eq__le__subst,axiom,
% 0.31/0.47 ! [A: real,F: real > real,B: real,C: real] :
% 0.31/0.47 ( ( A
% 0.31/0.47 = ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_subst
% 0.31/0.47 thf(fact_241_ord__eq__le__subst,axiom,
% 0.31/0.47 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.31/0.47 ( ( A
% 0.31/0.47 = ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_subst
% 0.31/0.47 thf(fact_242_ord__eq__le__subst,axiom,
% 0.31/0.47 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.31/0.47 ( ( A
% 0.31/0.47 = ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_subst
% 0.31/0.47 thf(fact_243_ord__eq__le__subst,axiom,
% 0.31/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.47 ( ( A
% 0.31/0.47 = ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_subst
% 0.31/0.47 thf(fact_244_order__subst2,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.31/0.47 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst2
% 0.31/0.47 thf(fact_245_order__subst2,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.31/0.47 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst2
% 0.31/0.47 thf(fact_246_order__subst2,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.47 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst2
% 0.31/0.47 thf(fact_247_order__subst2,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.47 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst2
% 0.31/0.47 thf(fact_248_order__subst1,axiom,
% 0.31/0.47 ! [A: real,F: real > real,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst1
% 0.31/0.47 thf(fact_249_order__subst1,axiom,
% 0.31/0.47 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst1
% 0.31/0.47 thf(fact_250_order__subst1,axiom,
% 0.31/0.47 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst1
% 0.31/0.47 thf(fact_251_order__subst1,axiom,
% 0.31/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst1
% 0.31/0.47 thf(fact_252_ord__eq__less__subst,axiom,
% 0.31/0.47 ! [A: real,F: real > real,B: real,C: real] :
% 0.31/0.47 ( ( A
% 0.31/0.47 = ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_less_subst
% 0.31/0.47 thf(fact_253_ord__eq__less__subst,axiom,
% 0.31/0.47 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.31/0.47 ( ( A
% 0.31/0.47 = ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_less_subst
% 0.31/0.47 thf(fact_254_ord__eq__less__subst,axiom,
% 0.31/0.47 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.31/0.47 ( ( A
% 0.31/0.47 = ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_less_subst
% 0.31/0.47 thf(fact_255_ord__eq__less__subst,axiom,
% 0.31/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.47 ( ( A
% 0.31/0.47 = ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_less_subst
% 0.31/0.47 thf(fact_256_ord__less__eq__subst,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > real,C: real] :
% 0.31/0.47 ( ( ord_less_real @ A @ B )
% 0.31/0.47 => ( ( ( F @ B )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_less_eq_subst
% 0.31/0.47 thf(fact_257_ord__less__eq__subst,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_real @ A @ B )
% 0.31/0.47 => ( ( ( F @ B )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_less_eq_subst
% 0.31/0.47 thf(fact_258_ord__less__eq__subst,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.31/0.47 ( ( ord_less_nat @ A @ B )
% 0.31/0.47 => ( ( ( F @ B )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_less_eq_subst
% 0.31/0.47 thf(fact_259_ord__less__eq__subst,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A @ B )
% 0.31/0.47 => ( ( ( F @ B )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_less_eq_subst
% 0.31/0.47 thf(fact_260_order__less__subst1,axiom,
% 0.31/0.47 ! [A: real,F: real > real,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst1
% 0.31/0.47 thf(fact_261_order__less__subst1,axiom,
% 0.31/0.47 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst1
% 0.31/0.47 thf(fact_262_order__less__subst1,axiom,
% 0.31/0.47 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst1
% 0.31/0.47 thf(fact_263_order__less__subst1,axiom,
% 0.31/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst1
% 0.31/0.47 thf(fact_264_order__less__subst2,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > real,C: real] :
% 0.31/0.47 ( ( ord_less_real @ A @ B )
% 0.31/0.47 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst2
% 0.31/0.47 thf(fact_265_order__less__subst2,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_real @ A @ B )
% 0.31/0.47 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst2
% 0.31/0.47 thf(fact_266_order__less__subst2,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.31/0.47 ( ( ord_less_nat @ A @ B )
% 0.31/0.47 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst2
% 0.31/0.47 thf(fact_267_order__less__subst2,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A @ B )
% 0.31/0.47 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst2
% 0.31/0.47 thf(fact_268_lt__ex,axiom,
% 0.31/0.47 ! [X2: real] :
% 0.31/0.47 ? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% 0.31/0.47
% 0.31/0.47 % lt_ex
% 0.31/0.47 thf(fact_269_gt__ex,axiom,
% 0.31/0.47 ! [X2: real] :
% 0.31/0.47 ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 0.31/0.47
% 0.31/0.47 % gt_ex
% 0.31/0.47 thf(fact_270_gt__ex,axiom,
% 0.31/0.47 ! [X2: nat] :
% 0.31/0.47 ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% 0.31/0.47
% 0.31/0.47 % gt_ex
% 0.31/0.47 thf(fact_271_neqE,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( X2 != Y )
% 0.31/0.47 => ( ~ ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neqE
% 0.31/0.47 thf(fact_272_neqE,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( X2 != Y )
% 0.31/0.47 => ( ~ ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neqE
% 0.31/0.47 thf(fact_273_neq__iff,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( X2 != Y )
% 0.31/0.47 = ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 | ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neq_iff
% 0.31/0.47 thf(fact_274_neq__iff,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( X2 != Y )
% 0.31/0.47 = ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 | ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neq_iff
% 0.31/0.47 thf(fact_275_order_Oasym,axiom,
% 0.31/0.47 ! [A: real,B: real] :
% 0.31/0.47 ( ( ord_less_real @ A @ B )
% 0.31/0.47 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.asym
% 0.31/0.47 thf(fact_276_order_Oasym,axiom,
% 0.31/0.47 ! [A: nat,B: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A @ B )
% 0.31/0.47 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.asym
% 0.31/0.47 thf(fact_277_dense,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ? [Z5: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Z5 )
% 0.31/0.47 & ( ord_less_real @ Z5 @ Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense
% 0.31/0.47 thf(fact_278_less__imp__neq,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ( X2 != Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_neq
% 0.31/0.47 thf(fact_279_less__imp__neq,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ( X2 != Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_neq
% 0.31/0.47 thf(fact_280_less__asym,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_asym
% 0.31/0.47 thf(fact_281_less__asym,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_asym
% 0.31/0.47 thf(fact_282_less__asym_H,axiom,
% 0.31/0.47 ! [A: real,B: real] :
% 0.31/0.47 ( ( ord_less_real @ A @ B )
% 0.31/0.47 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_asym'
% 0.31/0.47 thf(fact_283_less__asym_H,axiom,
% 0.31/0.47 ! [A: nat,B: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A @ B )
% 0.31/0.47 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_asym'
% 0.31/0.47 thf(fact_284_less__trans,axiom,
% 0.31/0.47 ! [X2: real,Y: real,Z: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ( ( ord_less_real @ Y @ Z )
% 0.31/0.47 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_trans
% 0.31/0.47 thf(fact_285_less__trans,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat,Z: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ( ( ord_less_nat @ Y @ Z )
% 0.31/0.47 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_trans
% 0.31/0.47 thf(fact_286_less__linear,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 | ( X2 = Y )
% 0.31/0.47 | ( ord_less_real @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_linear
% 0.31/0.47 thf(fact_287_less__linear,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 | ( X2 = Y )
% 0.31/0.47 | ( ord_less_nat @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_linear
% 0.31/0.47 thf(fact_288_less__irrefl,axiom,
% 0.31/0.47 ! [X2: real] :
% 0.31/0.47 ~ ( ord_less_real @ X2 @ X2 ) ).
% 0.31/0.47
% 0.31/0.47 % less_irrefl
% 0.31/0.47 thf(fact_289_less__irrefl,axiom,
% 0.31/0.47 ! [X2: nat] :
% 0.31/0.47 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 0.31/0.47
% 0.31/0.47 % less_irrefl
% 0.31/0.47 thf(fact_290_ord__eq__less__trans,axiom,
% 0.31/0.47 ! [A: real,B: real,C: real] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( ( ord_less_real @ B @ C )
% 0.31/0.47 => ( ord_less_real @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_less_trans
% 0.31/0.47 thf(fact_291_ord__eq__less__trans,axiom,
% 0.31/0.47 ! [A: nat,B: nat,C: nat] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( ( ord_less_nat @ B @ C )
% 0.31/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_less_trans
% 0.31/0.47 thf(fact_292_ord__less__eq__trans,axiom,
% 0.31/0.47 ! [A: real,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_real @ A @ B )
% 0.31/0.47 => ( ( B = C )
% 0.31/0.47 => ( ord_less_real @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_less_eq_trans
% 0.31/0.47 thf(fact_293_ord__less__eq__trans,axiom,
% 0.31/0.47 ! [A: nat,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A @ B )
% 0.31/0.47 => ( ( B = C )
% 0.31/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_less_eq_trans
% 0.31/0.47 thf(fact_294_dual__order_Oasym,axiom,
% 0.31/0.47 ! [B: real,A: real] :
% 0.31/0.47 ( ( ord_less_real @ B @ A )
% 0.31/0.47 => ~ ( ord_less_real @ A @ B ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.asym
% 0.31/0.47 thf(fact_295_dual__order_Oasym,axiom,
% 0.31/0.47 ! [B: nat,A: nat] :
% 0.31/0.47 ( ( ord_less_nat @ B @ A )
% 0.31/0.47 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.asym
% 0.31/0.47 thf(fact_296_less__imp__not__eq,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ( X2 != Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_not_eq
% 0.31/0.47 thf(fact_297_less__imp__not__eq,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ( X2 != Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_not_eq
% 0.31/0.47 thf(fact_298_less__not__sym,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_not_sym
% 0.31/0.47 thf(fact_299_less__not__sym,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_not_sym
% 0.31/0.47 thf(fact_300_less__induct,axiom,
% 0.31/0.47 ! [P3: nat > $o,A: nat] :
% 0.31/0.47 ( ! [X3: nat] :
% 0.31/0.47 ( ! [Y4: nat] :
% 0.31/0.47 ( ( ord_less_nat @ Y4 @ X3 )
% 0.31/0.47 => ( P3 @ Y4 ) )
% 0.31/0.47 => ( P3 @ X3 ) )
% 0.31/0.47 => ( P3 @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_induct
% 0.31/0.47 thf(fact_301_antisym__conv3,axiom,
% 0.31/0.47 ! [Y: real,X2: real] :
% 0.31/0.47 ( ~ ( ord_less_real @ Y @ X2 )
% 0.31/0.47 => ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 0.31/0.47 = ( X2 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv3
% 0.31/0.47 thf(fact_302_antisym__conv3,axiom,
% 0.31/0.47 ! [Y: nat,X2: nat] :
% 0.31/0.47 ( ~ ( ord_less_nat @ Y @ X2 )
% 0.31/0.47 => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 0.31/0.47 = ( X2 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv3
% 0.31/0.47 thf(fact_303_less__imp__not__eq2,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ( Y != X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_not_eq2
% 0.31/0.47 thf(fact_304_less__imp__not__eq2,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ( Y != X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_not_eq2
% 0.31/0.47 thf(fact_305_less__imp__triv,axiom,
% 0.31/0.47 ! [X2: real,Y: real,P3: $o] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ( ( ord_less_real @ Y @ X2 )
% 0.31/0.47 => P3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_triv
% 0.31/0.47 thf(fact_306_less__imp__triv,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat,P3: $o] :
% 0.31/0.47 ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ( ( ord_less_nat @ Y @ X2 )
% 0.31/0.47 => P3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_triv
% 0.31/0.47 thf(fact_307_linorder__cases,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ~ ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ( ( X2 != Y )
% 0.31/0.47 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_cases
% 0.31/0.47 thf(fact_308_linorder__cases,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ~ ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ( ( X2 != Y )
% 0.31/0.47 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_cases
% 0.31/0.47 thf(fact_309_dual__order_Oirrefl,axiom,
% 0.31/0.47 ! [A: real] :
% 0.31/0.47 ~ ( ord_less_real @ A @ A ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.irrefl
% 0.31/0.47 thf(fact_310_dual__order_Oirrefl,axiom,
% 0.31/0.47 ! [A: nat] :
% 0.31/0.47 ~ ( ord_less_nat @ A @ A ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.irrefl
% 0.31/0.47 thf(fact_311_order_Ostrict__trans,axiom,
% 0.31/0.47 ! [A: real,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_real @ A @ B )
% 0.31/0.47 => ( ( ord_less_real @ B @ C )
% 0.31/0.47 => ( ord_less_real @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_trans
% 0.31/0.47 thf(fact_312_order_Ostrict__trans,axiom,
% 0.31/0.47 ! [A: nat,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A @ B )
% 0.31/0.47 => ( ( ord_less_nat @ B @ C )
% 0.31/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_trans
% 0.31/0.47 thf(fact_313_less__imp__not__less,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_not_less
% 0.31/0.47 thf(fact_314_less__imp__not__less,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_not_less
% 0.31/0.47 thf(fact_315_exists__least__iff,axiom,
% 0.31/0.47 ( ( ^ [P4: nat > $o] :
% 0.31/0.47 ? [X5: nat] : ( P4 @ X5 ) )
% 0.31/0.47 = ( ^ [P5: nat > $o] :
% 0.31/0.47 ? [N2: nat] :
% 0.31/0.47 ( ( P5 @ N2 )
% 0.31/0.47 & ! [M2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.31/0.47 => ~ ( P5 @ M2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % exists_least_iff
% 0.31/0.47 thf(fact_316_linorder__less__wlog,axiom,
% 0.31/0.47 ! [P3: real > real > $o,A: real,B: real] :
% 0.31/0.47 ( ! [A4: real,B3: real] :
% 0.31/0.47 ( ( ord_less_real @ A4 @ B3 )
% 0.31/0.47 => ( P3 @ A4 @ B3 ) )
% 0.31/0.47 => ( ! [A4: real] : ( P3 @ A4 @ A4 )
% 0.31/0.47 => ( ! [A4: real,B3: real] :
% 0.31/0.47 ( ( P3 @ B3 @ A4 )
% 0.31/0.47 => ( P3 @ A4 @ B3 ) )
% 0.31/0.47 => ( P3 @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_less_wlog
% 0.31/0.47 thf(fact_317_linorder__less__wlog,axiom,
% 0.31/0.47 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 0.31/0.47 ( ! [A4: nat,B3: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A4 @ B3 )
% 0.31/0.47 => ( P3 @ A4 @ B3 ) )
% 0.31/0.47 => ( ! [A4: nat] : ( P3 @ A4 @ A4 )
% 0.31/0.47 => ( ! [A4: nat,B3: nat] :
% 0.31/0.47 ( ( P3 @ B3 @ A4 )
% 0.31/0.47 => ( P3 @ A4 @ B3 ) )
% 0.31/0.47 => ( P3 @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_less_wlog
% 0.31/0.47 thf(fact_318_dual__order_Ostrict__trans,axiom,
% 0.31/0.47 ! [B: real,A: real,C: real] :
% 0.31/0.47 ( ( ord_less_real @ B @ A )
% 0.31/0.47 => ( ( ord_less_real @ C @ B )
% 0.31/0.47 => ( ord_less_real @ C @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_trans
% 0.31/0.47 thf(fact_319_dual__order_Ostrict__trans,axiom,
% 0.31/0.47 ! [B: nat,A: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_nat @ B @ A )
% 0.31/0.47 => ( ( ord_less_nat @ C @ B )
% 0.31/0.47 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_trans
% 0.31/0.47 thf(fact_320_not__less__iff__gr__or__eq,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 0.31/0.47 = ( ( ord_less_real @ Y @ X2 )
% 0.31/0.47 | ( X2 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_less_iff_gr_or_eq
% 0.31/0.47 thf(fact_321_not__less__iff__gr__or__eq,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 0.31/0.47 = ( ( ord_less_nat @ Y @ X2 )
% 0.31/0.47 | ( X2 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_less_iff_gr_or_eq
% 0.31/0.47 thf(fact_322_order_Ostrict__implies__not__eq,axiom,
% 0.31/0.47 ! [A: real,B: real] :
% 0.31/0.47 ( ( ord_less_real @ A @ B )
% 0.31/0.47 => ( A != B ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_implies_not_eq
% 0.31/0.47 thf(fact_323_order_Ostrict__implies__not__eq,axiom,
% 0.31/0.47 ! [A: nat,B: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A @ B )
% 0.31/0.47 => ( A != B ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_implies_not_eq
% 0.31/0.47 thf(fact_324_dual__order_Ostrict__implies__not__eq,axiom,
% 0.31/0.47 ! [B: real,A: real] :
% 0.31/0.47 ( ( ord_less_real @ B @ A )
% 0.31/0.47 => ( A != B ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_implies_not_eq
% 0.31/0.47 thf(fact_325_dual__order_Ostrict__implies__not__eq,axiom,
% 0.31/0.47 ! [B: nat,A: nat] :
% 0.31/0.47 ( ( ord_less_nat @ B @ A )
% 0.31/0.47 => ( A != B ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_implies_not_eq
% 0.31/0.47 thf(fact_326_leD,axiom,
% 0.31/0.47 ! [Y: real,X2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ Y @ X2 )
% 0.31/0.47 => ~ ( ord_less_real @ X2 @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % leD
% 0.31/0.47 thf(fact_327_leD,axiom,
% 0.31/0.47 ! [Y: nat,X2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ Y @ X2 )
% 0.31/0.47 => ~ ( ord_less_nat @ X2 @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % leD
% 0.31/0.47 thf(fact_328_leI,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ~ ( ord_less_real @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % leI
% 0.31/0.47 thf(fact_329_leI,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ~ ( ord_less_nat @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % leI
% 0.31/0.47 thf(fact_330_le__less,axiom,
% 0.31/0.47 ( ord_less_eq_real
% 0.31/0.47 = ( ^ [X4: real,Y3: real] :
% 0.31/0.47 ( ( ord_less_real @ X4 @ Y3 )
% 0.31/0.47 | ( X4 = Y3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_less
% 0.31/0.47 thf(fact_331_le__less,axiom,
% 0.31/0.47 ( ord_less_eq_nat
% 0.31/0.47 = ( ^ [X4: nat,Y3: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X4 @ Y3 )
% 0.31/0.47 | ( X4 = Y3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_less
% 0.31/0.47 thf(fact_332_less__le,axiom,
% 0.31/0.47 ( ord_less_real
% 0.31/0.47 = ( ^ [X4: real,Y3: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.31/0.47 & ( X4 != Y3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_le
% 0.31/0.47 thf(fact_333_less__le,axiom,
% 0.31/0.47 ( ord_less_nat
% 0.31/0.47 = ( ^ [X4: nat,Y3: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.31/0.47 & ( X4 != Y3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_le
% 0.31/0.47 thf(fact_334_order__le__less__subst1,axiom,
% 0.31/0.47 ! [A: real,F: real > real,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst1
% 0.31/0.47 thf(fact_335_order__le__less__subst1,axiom,
% 0.31/0.47 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst1
% 0.31/0.47 thf(fact_336_order__le__less__subst1,axiom,
% 0.31/0.47 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst1
% 0.31/0.47 thf(fact_337_order__le__less__subst1,axiom,
% 0.31/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst1
% 0.31/0.47 thf(fact_338_order__le__less__subst2,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.31/0.47 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst2
% 0.31/0.47 thf(fact_339_order__le__less__subst2,axiom,
% 0.31/0.47 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.31/0.47 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst2
% 0.31/0.47 thf(fact_340_order__le__less__subst2,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.47 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst2
% 0.31/0.47 thf(fact_341_order__le__less__subst2,axiom,
% 0.31/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.47 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst2
% 0.31/0.47 thf(fact_342_order__less__le__subst1,axiom,
% 0.31/0.47 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.31/0.47 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.31/0.47 => ( ! [X3: real,Y2: real] :
% 0.31/0.47 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_le_subst1
% 0.31/0.47 thf(fact_343_order__less__le__subst1,axiom,
% 0.31/0.47 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_le_subst1
% 0.31/0.47 thf(fact_344_order__less__le__subst1,axiom,
% 0.31/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.47 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.31/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.47 => ( ! [X3: nat,Y2: nat] :
% 0.31/0.47 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_le_subst1
% 0.31/0.47
% 0.31/0.47 % Helper facts (5)
% 0.31/0.47 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( if_nat @ $false @ X2 @ Y )
% 0.31/0.47 = Y ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.31/0.47 ! [X2: nat,Y: nat] :
% 0.31/0.47 ( ( if_nat @ $true @ X2 @ Y )
% 0.31/0.47 = X2 ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
% 0.31/0.47 ! [P3: $o] :
% 0.31/0.47 ( ( P3 = $true )
% 0.31/0.47 | ( P3 = $false ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( if_real @ $false @ X2 @ Y )
% 0.31/0.47 = Y ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.31/0.47 ! [X2: real,Y: real] :
% 0.31/0.47 ( ( if_real @ $true @ X2 @ Y )
% 0.31/0.47 = X2 ) ).
% 0.31/0.47
% 0.31/0.47 % Conjectures (1)
% 0.31/0.47 thf(conj_0,conjecture,
% 0.31/0.47 ! [X3: real] :
% 0.31/0.47 ( ( ( poly_real2 @ p @ X3 )
% 0.31/0.47 != zero_zero_real )
% 0.31/0.47 | ( ord_less_real @ lb @ X3 ) ) ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 0.32/1.60 ---- Embedded in THF ---
% 0.32/1.60 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.32/1.60 %%% Generated on Tue Feb 24 08:29:32 EST 2026
% 0.32/1.60 %%% using '$$dhol' embedding, version 1.3.0.
% 0.32/1.60 %%% Logic specification used:
% 0.32/1.60 %%% thf(spec, logic, $$dhol).
% 0.32/1.60
% 0.32/1.60 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.iTAcLxWr78/DTF2DTF18417.p
% See solution above
% 0.32/1.61 ------------------------
% 0.32/1.63 ---- Cleaned THF ---
% 0.32/1.63 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 0.32/1.63 poly_poly_poly_real: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.32/1.63 poly_poly_real: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 0.32/1.63 poly_poly_nat: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 poly_real: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.32/1.63 poly_nat: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.32/1.63 set_real: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_n_t__Real__Oreal,type,
% 0.32/1.63 real: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(ty_n_t__Nat__Onat,type,
% 0.32/1.63 nat: $tType ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Osgn__class_Osgn_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.32/1.63 sgn_sg2128174761y_real: poly_poly_real > poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Osgn__class_Osgn_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 sgn_sgn_poly_real: poly_real > poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
% 0.32/1.63 sgn_sgn_real: real > real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.32/1.63 zero_zero_nat: nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.32/1.63 zero_zero_poly_nat: poly_nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 0.32/1.63 zero_z1059985641ly_nat: poly_poly_nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 0.32/1.63 zero_z935034829y_real: poly_poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.32/1.63 zero_z1423781445y_real: poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 zero_zero_poly_real: poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.32/1.63 zero_zero_real: real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_If_001t__Nat__Onat,type,
% 0.32/1.63 if_nat: $o > nat > nat > nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_If_001t__Real__Oreal,type,
% 0.32/1.63 if_real: $o > real > real > real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.32/1.63 ord_less_nat: nat > nat > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.32/1.63 ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 ord_less_poly_real: poly_real > poly_real > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.32/1.63 ord_less_real: real > real > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.32/1.63 ord_less_eq_nat: nat > nat > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.32/1.63 ord_le893774876y_real: poly_poly_real > poly_poly_real > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 ord_le1180086932y_real: poly_real > poly_real > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.32/1.63 ord_less_eq_real: real > real > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
% 0.32/1.63 ord_min_nat: nat > nat > nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Orderings_Oord__class_Omin_001t__Real__Oreal,type,
% 0.32/1.63 ord_min_real: real > real > real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.32/1.63 divide924636027y_real: poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > nat > nat > poly_poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Odivide__poly__main_001t__Real__Oreal,type,
% 0.32/1.63 divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat,type,
% 0.32/1.63 is_zero_nat: poly_nat > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 is_zero_poly_real: poly_poly_real > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Ois__zero_001t__Real__Oreal,type,
% 0.32/1.63 is_zero_real: poly_real > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.32/1.63 order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 order_poly_real: poly_real > poly_poly_real > nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Oorder_001t__Real__Oreal,type,
% 0.32/1.63 order_real: real > poly_real > nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
% 0.32/1.63 poly_nat2: poly_nat > nat > nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.32/1.63 poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.32/1.63 poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 poly_poly_real2: poly_poly_real > poly_real > poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
% 0.32/1.63 poly_real2: poly_real > real > real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat,type,
% 0.32/1.63 poly_cutoff_nat: nat > poly_nat > poly_nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly__cutoff_001t__Real__Oreal,type,
% 0.32/1.63 poly_cutoff_real: nat > poly_real > poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat,type,
% 0.32/1.63 poly_shift_nat: nat > poly_nat > poly_nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Opoly__shift_001t__Real__Oreal,type,
% 0.32/1.63 poly_shift_real: nat > poly_real > poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
% 0.32/1.63 reflect_poly_nat: poly_nat > poly_nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.32/1.63 reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.32/1.63 reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.32/1.63 reflec1522834046y_real: poly_poly_real > poly_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal,type,
% 0.32/1.63 reflect_poly_real: poly_real > poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.32/1.63 collect_real: ( real > $o ) > set_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Osgn__neg__inf_001t__Real__Oreal,type,
% 0.32/1.63 sturm_1076696862f_real: poly_real > real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Osgn__pos__inf_001t__Real__Oreal,type,
% 0.32/1.63 sturm_1308388506f_real: poly_real > real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_c_member_001t__Real__Oreal,type,
% 0.32/1.63 member_real: real > set_real > $o ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_lb1____,type,
% 0.32/1.63 lb1: real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_lb2____,type,
% 0.32/1.63 lb2: real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_lb____,type,
% 0.32/1.63 lb: real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_p,type,
% 0.32/1.63 p: poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(sy_v_thesis,type,
% 0.32/1.63 thesis: $o ).
% 0.32/1.63
% 0.32/1.63 thf(fact_0_lb1,axiom,
% 0.32/1.63 ! [X: real] :
% 0.32/1.63 ( ( ( poly_real2 @ p @ X )
% 0.32/1.63 = zero_zero_real )
% 0.32/1.63 => ( ord_less_real @ lb1 @ X ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_1_lb__def,axiom,
% 0.32/1.63 ( lb
% 0.32/1.63 = ( ord_min_real @ lb1 @ lb2 ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb1_O_A_092_060forall_062x_O_Apoly_Ap_Ax_A_061_A0_A_092_060longrightarrow_062_Alb1_A_060_Ax_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.32/1.63 ~ ! [Lb1: real] :
% 0.32/1.63 ~ ! [X: real] :
% 0.32/1.63 ( ( ( poly_real2 @ p @ X )
% 0.32/1.63 = zero_zero_real )
% 0.32/1.63 => ( ord_less_real @ Lb1 @ X ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_3_assms,axiom,
% 0.32/1.63 p != zero_zero_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(fact_4_poly__0,axiom,
% 0.32/1.63 ! [X2: poly_nat] :
% 0.32/1.63 ( ( poly_poly_nat2 @ zero_z1059985641ly_nat @ X2 )
% 0.32/1.63 = zero_zero_poly_nat ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_5_poly__0,axiom,
% 0.32/1.63 ! [X2: poly_poly_real] :
% 0.32/1.63 ( ( poly_poly_poly_real2 @ zero_z935034829y_real @ X2 )
% 0.32/1.63 = zero_z1423781445y_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_6_poly__0,axiom,
% 0.32/1.63 ! [X2: poly_real] :
% 0.32/1.63 ( ( poly_poly_real2 @ zero_z1423781445y_real @ X2 )
% 0.32/1.63 = zero_zero_poly_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_7_poly__0,axiom,
% 0.32/1.63 ! [X2: nat] :
% 0.32/1.63 ( ( poly_nat2 @ zero_zero_poly_nat @ X2 )
% 0.32/1.63 = zero_zero_nat ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_8_poly__0,axiom,
% 0.32/1.63 ! [X2: real] :
% 0.32/1.63 ( ( poly_real2 @ zero_zero_poly_real @ X2 )
% 0.32/1.63 = zero_zero_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_9_not__gr__zero,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.32/1.63 = ( N = zero_zero_nat ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_10_poly__IVT__neg,axiom,
% 0.32/1.63 ! [A: real,B: real,P: poly_real] :
% 0.32/1.63 ( ( ord_less_real @ A @ B )
% 0.32/1.63 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A ) )
% 0.32/1.63 => ( ( ord_less_real @ ( poly_real2 @ P @ B ) @ zero_zero_real )
% 0.32/1.63 => ? [X3: real] :
% 0.32/1.63 ( ( ord_less_real @ A @ X3 )
% 0.32/1.63 & ( ord_less_real @ X3 @ B )
% 0.32/1.63 & ( ( poly_real2 @ P @ X3 )
% 0.32/1.63 = zero_zero_real ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_11_poly__IVT__pos,axiom,
% 0.32/1.63 ! [A: real,B: real,P: poly_real] :
% 0.32/1.63 ( ( ord_less_real @ A @ B )
% 0.32/1.63 => ( ( ord_less_real @ ( poly_real2 @ P @ A ) @ zero_zero_real )
% 0.32/1.63 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B ) )
% 0.32/1.63 => ? [X3: real] :
% 0.32/1.63 ( ( ord_less_real @ A @ X3 )
% 0.32/1.63 & ( ord_less_real @ X3 @ B )
% 0.32/1.63 & ( ( poly_real2 @ P @ X3 )
% 0.32/1.63 = zero_zero_real ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_12_poly__all__0__iff__0,axiom,
% 0.32/1.63 ! [P: poly_poly_poly_real] :
% 0.32/1.63 ( ( ! [X4: poly_poly_real] :
% 0.32/1.63 ( ( poly_poly_poly_real2 @ P @ X4 )
% 0.32/1.63 = zero_z1423781445y_real ) )
% 0.32/1.63 = ( P = zero_z935034829y_real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_13_poly__all__0__iff__0,axiom,
% 0.32/1.63 ! [P: poly_real] :
% 0.32/1.63 ( ( ! [X4: real] :
% 0.32/1.63 ( ( poly_real2 @ P @ X4 )
% 0.32/1.63 = zero_zero_real ) )
% 0.32/1.63 = ( P = zero_zero_poly_real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_14_poly__all__0__iff__0,axiom,
% 0.32/1.63 ! [P: poly_poly_real] :
% 0.32/1.63 ( ( ! [X4: poly_real] :
% 0.32/1.63 ( ( poly_poly_real2 @ P @ X4 )
% 0.32/1.63 = zero_zero_poly_real ) )
% 0.32/1.63 = ( P = zero_z1423781445y_real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_15_less__numeral__extra_I3_J,axiom,
% 0.32/1.63 ~ ( ord_less_poly_real @ zero_zero_poly_real @ zero_zero_poly_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_16_less__numeral__extra_I3_J,axiom,
% 0.32/1.63 ~ ( ord_le38482960y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_17_less__numeral__extra_I3_J,axiom,
% 0.32/1.63 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_18_less__numeral__extra_I3_J,axiom,
% 0.32/1.63 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_19_field__lbound__gt__zero,axiom,
% 0.32/1.63 ! [D1: real,D2: real] :
% 0.32/1.63 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.32/1.63 => ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.32/1.63 => ? [E: real] :
% 0.32/1.63 ( ( ord_less_real @ zero_zero_real @ E )
% 0.32/1.63 & ( ord_less_real @ E @ D1 )
% 0.32/1.63 & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_20_gr__zeroI,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( N != zero_zero_nat )
% 0.32/1.63 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_21_not__less__zero,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_22_gr__implies__not__zero,axiom,
% 0.32/1.63 ! [M: nat,N: nat] :
% 0.32/1.63 ( ( ord_less_nat @ M @ N )
% 0.32/1.63 => ( N != zero_zero_nat ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_23_zero__less__iff__neq__zero,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/1.63 = ( N != zero_zero_nat ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_24_zero__reorient,axiom,
% 0.32/1.63 ! [X2: real] :
% 0.32/1.63 ( ( zero_zero_real = X2 )
% 0.32/1.63 = ( X2 = zero_zero_real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_25_zero__reorient,axiom,
% 0.32/1.63 ! [X2: poly_real] :
% 0.32/1.63 ( ( zero_zero_poly_real = X2 )
% 0.32/1.63 = ( X2 = zero_zero_poly_real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_26_zero__reorient,axiom,
% 0.32/1.63 ! [X2: nat] :
% 0.32/1.63 ( ( zero_zero_nat = X2 )
% 0.32/1.63 = ( X2 = zero_zero_nat ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_27_zero__reorient,axiom,
% 0.32/1.63 ! [X2: poly_nat] :
% 0.32/1.63 ( ( zero_zero_poly_nat = X2 )
% 0.32/1.63 = ( X2 = zero_zero_poly_nat ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_28_zero__reorient,axiom,
% 0.32/1.63 ! [X2: poly_poly_real] :
% 0.32/1.63 ( ( zero_z1423781445y_real = X2 )
% 0.32/1.63 = ( X2 = zero_z1423781445y_real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_29_poly__eq__poly__eq__iff,axiom,
% 0.32/1.63 ! [P: poly_real,Q: poly_real] :
% 0.32/1.63 ( ( ( poly_real2 @ P )
% 0.32/1.63 = ( poly_real2 @ Q ) )
% 0.32/1.63 = ( P = Q ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_30_poly__eq__poly__eq__iff,axiom,
% 0.32/1.63 ! [P: poly_poly_real,Q: poly_poly_real] :
% 0.32/1.63 ( ( ( poly_poly_real2 @ P )
% 0.32/1.63 = ( poly_poly_real2 @ Q ) )
% 0.32/1.63 = ( P = Q ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_31_min__less__iff__conj,axiom,
% 0.32/1.63 ! [Z: real,X2: real,Y: real] :
% 0.32/1.63 ( ( ord_less_real @ Z @ ( ord_min_real @ X2 @ Y ) )
% 0.32/1.63 = ( ( ord_less_real @ Z @ X2 )
% 0.32/1.63 & ( ord_less_real @ Z @ Y ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_32_min__less__iff__conj,axiom,
% 0.32/1.63 ! [Z: nat,X2: nat,Y: nat] :
% 0.32/1.63 ( ( ord_less_nat @ Z @ ( ord_min_nat @ X2 @ Y ) )
% 0.32/1.63 = ( ( ord_less_nat @ Z @ X2 )
% 0.32/1.63 & ( ord_less_nat @ Z @ Y ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_33_min_Oidem,axiom,
% 0.32/1.63 ! [A: real] :
% 0.32/1.63 ( ( ord_min_real @ A @ A )
% 0.32/1.63 = A ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_34_min_Oidem,axiom,
% 0.32/1.63 ! [A: nat] :
% 0.32/1.63 ( ( ord_min_nat @ A @ A )
% 0.32/1.63 = A ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_35_min_Oleft__idem,axiom,
% 0.32/1.63 ! [A: real,B: real] :
% 0.32/1.63 ( ( ord_min_real @ A @ ( ord_min_real @ A @ B ) )
% 0.32/1.63 = ( ord_min_real @ A @ B ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_36_min_Oleft__idem,axiom,
% 0.32/1.63 ! [A: nat,B: nat] :
% 0.32/1.63 ( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
% 0.32/1.63 = ( ord_min_nat @ A @ B ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_37_min_Oright__idem,axiom,
% 0.32/1.63 ! [A: real,B: real] :
% 0.32/1.63 ( ( ord_min_real @ ( ord_min_real @ A @ B ) @ B )
% 0.32/1.63 = ( ord_min_real @ A @ B ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_38_min_Oright__idem,axiom,
% 0.32/1.63 ! [A: nat,B: nat] :
% 0.32/1.63 ( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
% 0.32/1.63 = ( ord_min_nat @ A @ B ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_39_min_Ostrict__coboundedI2,axiom,
% 0.32/1.63 ! [B: real,C: real,A: real] :
% 0.32/1.63 ( ( ord_less_real @ B @ C )
% 0.32/1.63 => ( ord_less_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_40_min_Ostrict__coboundedI2,axiom,
% 0.32/1.63 ! [B: nat,C: nat,A: nat] :
% 0.32/1.63 ( ( ord_less_nat @ B @ C )
% 0.32/1.63 => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_41_min_Ostrict__coboundedI1,axiom,
% 0.32/1.63 ! [A: real,C: real,B: real] :
% 0.32/1.63 ( ( ord_less_real @ A @ C )
% 0.32/1.63 => ( ord_less_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_42_min_Ostrict__coboundedI1,axiom,
% 0.32/1.63 ! [A: nat,C: nat,B: nat] :
% 0.32/1.63 ( ( ord_less_nat @ A @ C )
% 0.32/1.63 => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_43_min_Ostrict__order__iff,axiom,
% 0.32/1.63 ( ord_less_real
% 0.32/1.63 = ( ^ [A2: real,B2: real] :
% 0.32/1.63 ( ( A2
% 0.32/1.63 = ( ord_min_real @ A2 @ B2 ) )
% 0.32/1.63 & ( A2 != B2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_44_min_Ostrict__order__iff,axiom,
% 0.32/1.63 ( ord_less_nat
% 0.32/1.63 = ( ^ [A2: nat,B2: nat] :
% 0.32/1.63 ( ( A2
% 0.32/1.63 = ( ord_min_nat @ A2 @ B2 ) )
% 0.32/1.63 & ( A2 != B2 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_45_min_Ostrict__boundedE,axiom,
% 0.32/1.63 ! [A: real,B: real,C: real] :
% 0.32/1.63 ( ( ord_less_real @ A @ ( ord_min_real @ B @ C ) )
% 0.32/1.63 => ~ ( ( ord_less_real @ A @ B )
% 0.32/1.63 => ~ ( ord_less_real @ A @ C ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_46_min_Ostrict__boundedE,axiom,
% 0.32/1.63 ! [A: nat,B: nat,C: nat] :
% 0.32/1.63 ( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
% 0.32/1.63 => ~ ( ( ord_less_nat @ A @ B )
% 0.32/1.63 => ~ ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_47_min__less__iff__disj,axiom,
% 0.32/1.63 ! [X2: real,Y: real,Z: real] :
% 0.32/1.63 ( ( ord_less_real @ ( ord_min_real @ X2 @ Y ) @ Z )
% 0.32/1.63 = ( ( ord_less_real @ X2 @ Z )
% 0.32/1.63 | ( ord_less_real @ Y @ Z ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_48_min__less__iff__disj,axiom,
% 0.32/1.63 ! [X2: nat,Y: nat,Z: nat] :
% 0.32/1.63 ( ( ord_less_nat @ ( ord_min_nat @ X2 @ Y ) @ Z )
% 0.32/1.63 = ( ( ord_less_nat @ X2 @ Z )
% 0.32/1.63 | ( ord_less_nat @ Y @ Z ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_49_is__zero__null,axiom,
% 0.32/1.63 ( is_zero_real
% 0.32/1.63 = ( ^ [P2: poly_real] : ( P2 = zero_zero_poly_real ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_50_is__zero__null,axiom,
% 0.32/1.63 ( is_zero_nat
% 0.32/1.63 = ( ^ [P2: poly_nat] : ( P2 = zero_zero_poly_nat ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_51_is__zero__null,axiom,
% 0.32/1.63 ( is_zero_poly_real
% 0.32/1.63 = ( ^ [P2: poly_poly_real] : ( P2 = zero_z1423781445y_real ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_52_min_Oleft__commute,axiom,
% 0.32/1.63 ! [B: real,A: real,C: real] :
% 0.32/1.63 ( ( ord_min_real @ B @ ( ord_min_real @ A @ C ) )
% 0.32/1.63 = ( ord_min_real @ A @ ( ord_min_real @ B @ C ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_53_min_Oleft__commute,axiom,
% 0.32/1.63 ! [B: nat,A: nat,C: nat] :
% 0.32/1.63 ( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
% 0.32/1.63 = ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_54_min_Ocommute,axiom,
% 0.32/1.63 ( ord_min_real
% 0.32/1.63 = ( ^ [A2: real,B2: real] : ( ord_min_real @ B2 @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_55_min_Ocommute,axiom,
% 0.32/1.63 ( ord_min_nat
% 0.32/1.63 = ( ^ [A2: nat,B2: nat] : ( ord_min_nat @ B2 @ A2 ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_56_min_Oassoc,axiom,
% 0.32/1.63 ! [A: real,B: real,C: real] :
% 0.32/1.63 ( ( ord_min_real @ ( ord_min_real @ A @ B ) @ C )
% 0.32/1.63 = ( ord_min_real @ A @ ( ord_min_real @ B @ C ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_57_min_Oassoc,axiom,
% 0.32/1.63 ! [A: nat,B: nat,C: nat] :
% 0.32/1.63 ( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
% 0.32/1.63 = ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_58_poly__cutoff__0,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( poly_cutoff_real @ N @ zero_zero_poly_real )
% 0.32/1.63 = zero_zero_poly_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_59_poly__cutoff__0,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
% 0.32/1.63 = zero_zero_poly_nat ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_60_poly__cutoff__0,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( poly_c1404107022y_real @ N @ zero_z1423781445y_real )
% 0.32/1.63 = zero_z1423781445y_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_61_last__non__root__interval,axiom,
% 0.32/1.63 ! [P: poly_real,Ub: real] :
% 0.32/1.63 ( ( P != zero_zero_poly_real )
% 0.32/1.63 => ~ ! [Lb: real] :
% 0.32/1.63 ( ( ord_less_real @ Lb @ Ub )
% 0.32/1.63 => ~ ! [Z2: real] :
% 0.32/1.63 ( ( ( ord_less_eq_real @ Lb @ Z2 )
% 0.32/1.63 & ( ord_less_real @ Z2 @ Ub ) )
% 0.32/1.63 => ( ( poly_real2 @ P @ Z2 )
% 0.32/1.63 != zero_zero_real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_62_next__non__root__interval,axiom,
% 0.32/1.63 ! [P: poly_real,Lb2: real] :
% 0.32/1.63 ( ( P != zero_zero_poly_real )
% 0.32/1.63 => ~ ! [Ub2: real] :
% 0.32/1.63 ( ( ord_less_real @ Lb2 @ Ub2 )
% 0.32/1.63 => ~ ! [Z2: real] :
% 0.32/1.63 ( ( ( ord_less_real @ Lb2 @ Z2 )
% 0.32/1.63 & ( ord_less_eq_real @ Z2 @ Ub2 ) )
% 0.32/1.63 => ( ( poly_real2 @ P @ Z2 )
% 0.32/1.63 != zero_zero_real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_63_reflect__poly__at__0__eq__0__iff,axiom,
% 0.32/1.63 ! [P: poly_real] :
% 0.32/1.63 ( ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
% 0.32/1.63 = zero_zero_real )
% 0.32/1.63 = ( P = zero_zero_poly_real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_64_reflect__poly__at__0__eq__0__iff,axiom,
% 0.32/1.63 ! [P: poly_poly_real] :
% 0.32/1.63 ( ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
% 0.32/1.63 = zero_zero_poly_real )
% 0.32/1.63 = ( P = zero_z1423781445y_real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_65_reflect__poly__at__0__eq__0__iff,axiom,
% 0.32/1.63 ! [P: poly_nat] :
% 0.32/1.63 ( ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 0.32/1.63 = zero_zero_nat )
% 0.32/1.63 = ( P = zero_zero_poly_nat ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_66_reflect__poly__at__0__eq__0__iff,axiom,
% 0.32/1.63 ! [P: poly_poly_nat] :
% 0.32/1.63 ( ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
% 0.32/1.63 = zero_zero_poly_nat )
% 0.32/1.63 = ( P = zero_z1059985641ly_nat ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_67_reflect__poly__at__0__eq__0__iff,axiom,
% 0.32/1.63 ! [P: poly_poly_poly_real] :
% 0.32/1.63 ( ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
% 0.32/1.63 = zero_z1423781445y_real )
% 0.32/1.63 = ( P = zero_z935034829y_real ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_68_lb2,axiom,
% 0.32/1.63 ! [X: real] :
% 0.32/1.63 ( ( ord_less_eq_real @ X @ lb2 )
% 0.32/1.63 => ( ( sgn_sgn_real @ ( poly_real2 @ p @ X ) )
% 0.32/1.63 = ( sturm_1076696862f_real @ p ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_69_poly__shift__0,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( poly_shift_real @ N @ zero_zero_poly_real )
% 0.32/1.63 = zero_zero_poly_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_70_poly__shift__0,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( poly_shift_nat @ N @ zero_zero_poly_nat )
% 0.32/1.63 = zero_zero_poly_nat ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_71_poly__shift__0,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( poly_shift_poly_real @ N @ zero_z1423781445y_real )
% 0.32/1.63 = zero_z1423781445y_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_72_order__root,axiom,
% 0.32/1.63 ! [P: poly_real,A: real] :
% 0.32/1.63 ( ( ( poly_real2 @ P @ A )
% 0.32/1.63 = zero_zero_real )
% 0.32/1.63 = ( ( P = zero_zero_poly_real )
% 0.32/1.63 | ( ( order_real @ A @ P )
% 0.32/1.63 != zero_zero_nat ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_73_order__root,axiom,
% 0.32/1.63 ! [P: poly_poly_real,A: poly_real] :
% 0.32/1.63 ( ( ( poly_poly_real2 @ P @ A )
% 0.32/1.63 = zero_zero_poly_real )
% 0.32/1.63 = ( ( P = zero_z1423781445y_real )
% 0.32/1.63 | ( ( order_poly_real @ A @ P )
% 0.32/1.63 != zero_zero_nat ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_74_order__root,axiom,
% 0.32/1.63 ! [P: poly_poly_poly_real,A: poly_poly_real] :
% 0.32/1.63 ( ( ( poly_poly_poly_real2 @ P @ A )
% 0.32/1.63 = zero_z1423781445y_real )
% 0.32/1.63 = ( ( P = zero_z935034829y_real )
% 0.32/1.63 | ( ( order_poly_poly_real @ A @ P )
% 0.32/1.63 != zero_zero_nat ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_75_divide__poly__main__0,axiom,
% 0.32/1.63 ! [R: poly_real,D: poly_real,Dr: nat,N: nat] :
% 0.32/1.63 ( ( divide1561404011n_real @ zero_zero_real @ zero_zero_poly_real @ R @ D @ Dr @ N )
% 0.32/1.63 = zero_zero_poly_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_76_divide__poly__main__0,axiom,
% 0.32/1.63 ! [R: poly_poly_real,D: poly_poly_real,Dr: nat,N: nat] :
% 0.32/1.63 ( ( divide1142363123y_real @ zero_zero_poly_real @ zero_z1423781445y_real @ R @ D @ Dr @ N )
% 0.32/1.63 = zero_z1423781445y_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_77_divide__poly__main__0,axiom,
% 0.32/1.63 ! [R: poly_poly_poly_real,D: poly_poly_poly_real,Dr: nat,N: nat] :
% 0.32/1.63 ( ( divide924636027y_real @ zero_z1423781445y_real @ zero_z935034829y_real @ R @ D @ Dr @ N )
% 0.32/1.63 = zero_z935034829y_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_78_not__eq__pos__or__neg__iff__1,axiom,
% 0.32/1.63 ! [Lb2: real,Ub: real,P: poly_real] :
% 0.32/1.63 ( ( ! [Z3: real] :
% 0.32/1.63 ( ( ( ord_less_real @ Lb2 @ Z3 )
% 0.32/1.63 & ( ord_less_eq_real @ Z3 @ Ub ) )
% 0.32/1.63 => ( ( poly_real2 @ P @ Z3 )
% 0.32/1.63 != zero_zero_real ) ) )
% 0.32/1.63 = ( ! [Z3: real] :
% 0.32/1.63 ( ( ( ord_less_real @ Lb2 @ Z3 )
% 0.32/1.63 & ( ord_less_eq_real @ Z3 @ Ub ) )
% 0.32/1.63 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z3 ) ) )
% 0.32/1.63 | ! [Z3: real] :
% 0.32/1.63 ( ( ( ord_less_real @ Lb2 @ Z3 )
% 0.32/1.63 & ( ord_less_eq_real @ Z3 @ Ub ) )
% 0.32/1.63 => ( ord_less_real @ ( poly_real2 @ P @ Z3 ) @ zero_zero_real ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_79__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb2_O_A_092_060forall_062x_092_060le_062lb2_O_Asgn_A_Ipoly_Ap_Ax_J_A_061_Asgn__neg__inf_Ap_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.32/1.63 ~ ! [Lb22: real] :
% 0.32/1.63 ~ ! [X: real] :
% 0.32/1.63 ( ( ord_less_eq_real @ X @ Lb22 )
% 0.32/1.63 => ( ( sgn_sgn_real @ ( poly_real2 @ p @ X ) )
% 0.32/1.63 = ( sturm_1076696862f_real @ p ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_80_le__zero__eq,axiom,
% 0.32/1.63 ! [N: nat] :
% 0.32/1.63 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/1.63 = ( N = zero_zero_nat ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_81_min_Obounded__iff,axiom,
% 0.32/1.63 ! [A: real,B: real,C: real] :
% 0.32/1.63 ( ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) )
% 0.32/1.63 = ( ( ord_less_eq_real @ A @ B )
% 0.32/1.63 & ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_82_min_Obounded__iff,axiom,
% 0.32/1.63 ! [A: nat,B: nat,C: nat] :
% 0.32/1.63 ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
% 0.32/1.63 = ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.63 & ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_83_reflect__poly__0,axiom,
% 0.32/1.63 ( ( reflect_poly_real @ zero_zero_poly_real )
% 0.32/1.63 = zero_zero_poly_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_84_reflect__poly__0,axiom,
% 0.32/1.63 ( ( reflect_poly_nat @ zero_zero_poly_nat )
% 0.32/1.63 = zero_zero_poly_nat ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_85_reflect__poly__0,axiom,
% 0.32/1.63 ( ( reflec1522834046y_real @ zero_z1423781445y_real )
% 0.32/1.63 = zero_z1423781445y_real ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_86_mem__Collect__eq,axiom,
% 0.32/1.63 ! [A: real,P3: real > $o] :
% 0.32/1.63 ( ( member_real @ A @ ( collect_real @ P3 ) )
% 0.32/1.63 = ( P3 @ A ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_87_Collect__mem__eq,axiom,
% 0.32/1.63 ! [A3: set_real] :
% 0.32/1.63 ( ( collect_real
% 0.32/1.63 @ ^ [X4: real] : ( member_real @ X4 @ A3 ) )
% 0.32/1.63 = A3 ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_88_that,axiom,
% 0.32/1.63 ! [Lb2: real] :
% 0.32/1.63 ( ! [X3: real] :
% 0.32/1.63 ( ( ( poly_real2 @ p @ X3 )
% 0.32/1.63 = zero_zero_real )
% 0.32/1.63 => ( ord_less_real @ Lb2 @ X3 ) )
% 0.32/1.63 => ( ! [X3: real] :
% 0.32/1.63 ( ( ord_less_eq_real @ X3 @ Lb2 )
% 0.32/1.63 => ( ( sgn_sgn_real @ ( poly_real2 @ p @ X3 ) )
% 0.32/1.63 = ( sturm_1076696862f_real @ p ) ) )
% 0.32/1.63 => thesis ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_89_complete__real,axiom,
% 0.32/1.63 ! [S: set_real] :
% 0.32/1.63 ( ? [X: real] : ( member_real @ X @ S )
% 0.32/1.63 => ( ? [Z2: real] :
% 0.32/1.63 ! [X3: real] :
% 0.32/1.63 ( ( member_real @ X3 @ S )
% 0.32/1.63 => ( ord_less_eq_real @ X3 @ Z2 ) )
% 0.32/1.63 => ? [Y2: real] :
% 0.32/1.63 ( ! [X: real] :
% 0.32/1.63 ( ( member_real @ X @ S )
% 0.32/1.63 => ( ord_less_eq_real @ X @ Y2 ) )
% 0.32/1.63 & ! [Z2: real] :
% 0.32/1.63 ( ! [X3: real] :
% 0.32/1.63 ( ( member_real @ X3 @ S )
% 0.32/1.63 => ( ord_less_eq_real @ X3 @ Z2 ) )
% 0.32/1.63 => ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_90_zero__le,axiom,
% 0.32/1.63 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_91_le__numeral__extra_I3_J,axiom,
% 0.32/1.63 ord_le1180086932y_real @ zero_zero_poly_real @ zero_zero_poly_real ).
% 0.32/1.63
% 0.32/1.63 thf(fact_92_le__numeral__extra_I3_J,axiom,
% 0.32/1.63 ord_le893774876y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ).
% 0.32/1.63
% 0.32/1.63 thf(fact_93_le__numeral__extra_I3_J,axiom,
% 0.32/1.63 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.32/1.63
% 0.32/1.63 thf(fact_94_le__numeral__extra_I3_J,axiom,
% 0.32/1.63 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.32/1.63
% 0.32/1.63 thf(fact_95_less__eq__real__def,axiom,
% 0.32/1.63 ( ord_less_eq_real
% 0.32/1.63 = ( ^ [X4: real,Y3: real] :
% 0.32/1.63 ( ( ord_less_real @ X4 @ Y3 )
% 0.32/1.63 | ( X4 = Y3 ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_96_min_Omono,axiom,
% 0.32/1.63 ! [A: real,C: real,B: real,D: real] :
% 0.32/1.63 ( ( ord_less_eq_real @ A @ C )
% 0.32/1.63 => ( ( ord_less_eq_real @ B @ D )
% 0.32/1.63 => ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ ( ord_min_real @ C @ D ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_97_min_Omono,axiom,
% 0.32/1.63 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.32/1.63 ( ( ord_less_eq_nat @ A @ C )
% 0.32/1.63 => ( ( ord_less_eq_nat @ B @ D )
% 0.32/1.63 => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D ) ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_98_min_OorderE,axiom,
% 0.32/1.63 ! [A: real,B: real] :
% 0.32/1.63 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.63 => ( A
% 0.32/1.63 = ( ord_min_real @ A @ B ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_99_min_OorderE,axiom,
% 0.32/1.63 ! [A: nat,B: nat] :
% 0.32/1.63 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.63 => ( A
% 0.32/1.63 = ( ord_min_nat @ A @ B ) ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_100_min_OorderI,axiom,
% 0.32/1.63 ! [A: real,B: real] :
% 0.32/1.63 ( ( A
% 0.32/1.63 = ( ord_min_real @ A @ B ) )
% 0.32/1.63 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_101_min_OorderI,axiom,
% 0.32/1.63 ! [A: nat,B: nat] :
% 0.32/1.63 ( ( A
% 0.32/1.63 = ( ord_min_nat @ A @ B ) )
% 0.32/1.63 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_102_min_Oabsorb1,axiom,
% 0.32/1.63 ! [A: real,B: real] :
% 0.32/1.63 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.63 => ( ( ord_min_real @ A @ B )
% 0.32/1.63 = A ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_103_min_Oabsorb1,axiom,
% 0.32/1.63 ! [A: nat,B: nat] :
% 0.32/1.63 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.63 => ( ( ord_min_nat @ A @ B )
% 0.32/1.63 = A ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_104_min_Oabsorb2,axiom,
% 0.32/1.63 ! [B: real,A: real] :
% 0.32/1.63 ( ( ord_less_eq_real @ B @ A )
% 0.32/1.63 => ( ( ord_min_real @ A @ B )
% 0.32/1.63 = B ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_105_min_Oabsorb2,axiom,
% 0.32/1.63 ! [B: nat,A: nat] :
% 0.32/1.63 ( ( ord_less_eq_nat @ B @ A )
% 0.32/1.63 => ( ( ord_min_nat @ A @ B )
% 0.32/1.63 = B ) ) ).
% 0.32/1.63
% 0.32/1.63 thf(fact_106_min_OboundedE,axiom,
% 0.32/1.63 ! [A: real,B: real,C: real] :
% 0.32/1.63 ( ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) )
% 0.32/1.63 => ~ ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ~ ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_107_min_OboundedE,axiom,
% 0.32/1.64 ! [A: nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
% 0.32/1.64 => ~ ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_108_min_OboundedI,axiom,
% 0.32/1.64 ! [A: real,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_real @ A @ C )
% 0.32/1.64 => ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_109_min_OboundedI,axiom,
% 0.32/1.64 ! [A: nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_nat @ A @ C )
% 0.32/1.64 => ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_110_min_Oorder__iff,axiom,
% 0.32/1.64 ( ord_less_eq_real
% 0.32/1.64 = ( ^ [A2: real,B2: real] :
% 0.32/1.64 ( A2
% 0.32/1.64 = ( ord_min_real @ A2 @ B2 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_111_min_Oorder__iff,axiom,
% 0.32/1.64 ( ord_less_eq_nat
% 0.32/1.64 = ( ^ [A2: nat,B2: nat] :
% 0.32/1.64 ( A2
% 0.32/1.64 = ( ord_min_nat @ A2 @ B2 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_112_min_Ocobounded1,axiom,
% 0.32/1.64 ! [A: real,B: real] : ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ A ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_113_min_Ocobounded1,axiom,
% 0.32/1.64 ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_114_min_Ocobounded2,axiom,
% 0.32/1.64 ! [A: real,B: real] : ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ B ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_115_min_Ocobounded2,axiom,
% 0.32/1.64 ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_116_min_Oabsorb__iff1,axiom,
% 0.32/1.64 ( ord_less_eq_real
% 0.32/1.64 = ( ^ [A2: real,B2: real] :
% 0.32/1.64 ( ( ord_min_real @ A2 @ B2 )
% 0.32/1.64 = A2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_117_min_Oabsorb__iff1,axiom,
% 0.32/1.64 ( ord_less_eq_nat
% 0.32/1.64 = ( ^ [A2: nat,B2: nat] :
% 0.32/1.64 ( ( ord_min_nat @ A2 @ B2 )
% 0.32/1.64 = A2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_118_min_Oabsorb__iff2,axiom,
% 0.32/1.64 ( ord_less_eq_real
% 0.32/1.64 = ( ^ [B2: real,A2: real] :
% 0.32/1.64 ( ( ord_min_real @ A2 @ B2 )
% 0.32/1.64 = B2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_119_min_Oabsorb__iff2,axiom,
% 0.32/1.64 ( ord_less_eq_nat
% 0.32/1.64 = ( ^ [B2: nat,A2: nat] :
% 0.32/1.64 ( ( ord_min_nat @ A2 @ B2 )
% 0.32/1.64 = B2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_120_min_OcoboundedI1,axiom,
% 0.32/1.64 ! [A: real,C: real,B: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ C )
% 0.32/1.64 => ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_121_min_OcoboundedI1,axiom,
% 0.32/1.64 ! [A: nat,C: nat,B: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ C )
% 0.32/1.64 => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_122_min_OcoboundedI2,axiom,
% 0.32/1.64 ! [B: real,C: real,A: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ B @ C )
% 0.32/1.64 => ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_123_min_OcoboundedI2,axiom,
% 0.32/1.64 ! [B: nat,C: nat,A: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ B @ C )
% 0.32/1.64 => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_124_min__le__iff__disj,axiom,
% 0.32/1.64 ! [X2: real,Y: real,Z: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ ( ord_min_real @ X2 @ Y ) @ Z )
% 0.32/1.64 = ( ( ord_less_eq_real @ X2 @ Z )
% 0.32/1.64 | ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_125_min__le__iff__disj,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat,Z: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ ( ord_min_nat @ X2 @ Y ) @ Z )
% 0.32/1.64 = ( ( ord_less_eq_nat @ X2 @ Z )
% 0.32/1.64 | ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_126_order__0I,axiom,
% 0.32/1.64 ! [P: poly_real,A: real] :
% 0.32/1.64 ( ( ( poly_real2 @ P @ A )
% 0.32/1.64 != zero_zero_real )
% 0.32/1.64 => ( ( order_real @ A @ P )
% 0.32/1.64 = zero_zero_nat ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_127_order__0I,axiom,
% 0.32/1.64 ! [P: poly_poly_real,A: poly_real] :
% 0.32/1.64 ( ( ( poly_poly_real2 @ P @ A )
% 0.32/1.64 != zero_zero_poly_real )
% 0.32/1.64 => ( ( order_poly_real @ A @ P )
% 0.32/1.64 = zero_zero_nat ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_128_order__0I,axiom,
% 0.32/1.64 ! [P: poly_poly_poly_real,A: poly_poly_real] :
% 0.32/1.64 ( ( ( poly_poly_poly_real2 @ P @ A )
% 0.32/1.64 != zero_z1423781445y_real )
% 0.32/1.64 => ( ( order_poly_poly_real @ A @ P )
% 0.32/1.64 = zero_zero_nat ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_129_not__eq__pos__or__neg__iff__2,axiom,
% 0.32/1.64 ! [Lb2: real,Ub: real,P: poly_real] :
% 0.32/1.64 ( ( ! [Z3: real] :
% 0.32/1.64 ( ( ( ord_less_eq_real @ Lb2 @ Z3 )
% 0.32/1.64 & ( ord_less_real @ Z3 @ Ub ) )
% 0.32/1.64 => ( ( poly_real2 @ P @ Z3 )
% 0.32/1.64 != zero_zero_real ) ) )
% 0.32/1.64 = ( ! [Z3: real] :
% 0.32/1.64 ( ( ( ord_less_eq_real @ Lb2 @ Z3 )
% 0.32/1.64 & ( ord_less_real @ Z3 @ Ub ) )
% 0.32/1.64 => ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z3 ) ) )
% 0.32/1.64 | ! [Z3: real] :
% 0.32/1.64 ( ( ( ord_less_eq_real @ Lb2 @ Z3 )
% 0.32/1.64 & ( ord_less_real @ Z3 @ Ub ) )
% 0.32/1.64 => ( ord_less_real @ ( poly_real2 @ P @ Z3 ) @ zero_zero_real ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_130_zero__le__sgn__iff,axiom,
% 0.32/1.64 ! [X2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X2 ) )
% 0.32/1.64 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_131_sgn__le__0__iff,axiom,
% 0.32/1.64 ! [X2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X2 ) @ zero_zero_real )
% 0.32/1.64 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_132_sgn__greater,axiom,
% 0.32/1.64 ! [A: poly_real] :
% 0.32/1.64 ( ( ord_less_poly_real @ zero_zero_poly_real @ ( sgn_sgn_poly_real @ A ) )
% 0.32/1.64 = ( ord_less_poly_real @ zero_zero_poly_real @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_133_sgn__greater,axiom,
% 0.32/1.64 ! [A: poly_poly_real] :
% 0.32/1.64 ( ( ord_le38482960y_real @ zero_z1423781445y_real @ ( sgn_sg2128174761y_real @ A ) )
% 0.32/1.64 = ( ord_le38482960y_real @ zero_z1423781445y_real @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_134_sgn__greater,axiom,
% 0.32/1.64 ! [A: real] :
% 0.32/1.64 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 0.32/1.64 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_135_sgn__less,axiom,
% 0.32/1.64 ! [A: poly_real] :
% 0.32/1.64 ( ( ord_less_poly_real @ ( sgn_sgn_poly_real @ A ) @ zero_zero_poly_real )
% 0.32/1.64 = ( ord_less_poly_real @ A @ zero_zero_poly_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_136_sgn__less,axiom,
% 0.32/1.64 ! [A: poly_poly_real] :
% 0.32/1.64 ( ( ord_le38482960y_real @ ( sgn_sg2128174761y_real @ A ) @ zero_z1423781445y_real )
% 0.32/1.64 = ( ord_le38482960y_real @ A @ zero_z1423781445y_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_137_sgn__less,axiom,
% 0.32/1.64 ! [A: real] :
% 0.32/1.64 ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 0.32/1.64 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_138_root__ub,axiom,
% 0.32/1.64 ! [P: poly_real] :
% 0.32/1.64 ( ( P != zero_zero_poly_real )
% 0.32/1.64 => ~ ! [Ub2: real] :
% 0.32/1.64 ( ! [X: real] :
% 0.32/1.64 ( ( ( poly_real2 @ P @ X )
% 0.32/1.64 = zero_zero_real )
% 0.32/1.64 => ( ord_less_real @ X @ Ub2 ) )
% 0.32/1.64 => ~ ! [X: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ Ub2 @ X )
% 0.32/1.64 => ( ( sgn_sgn_real @ ( poly_real2 @ P @ X ) )
% 0.32/1.64 = ( sturm_1308388506f_real @ P ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_139_sgn__0,axiom,
% 0.32/1.64 ( ( sgn_sgn_real @ zero_zero_real )
% 0.32/1.64 = zero_zero_real ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_140_sgn__0,axiom,
% 0.32/1.64 ( ( sgn_sgn_poly_real @ zero_zero_poly_real )
% 0.32/1.64 = zero_zero_poly_real ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_141_sgn__0,axiom,
% 0.32/1.64 ( ( sgn_sg2128174761y_real @ zero_z1423781445y_real )
% 0.32/1.64 = zero_z1423781445y_real ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_142_sgn__zero,axiom,
% 0.32/1.64 ( ( sgn_sgn_real @ zero_zero_real )
% 0.32/1.64 = zero_zero_real ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_143_sgn__sgn,axiom,
% 0.32/1.64 ! [A: real] :
% 0.32/1.64 ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
% 0.32/1.64 = ( sgn_sgn_real @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_144_linorder__neqE__linordered__idom,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( X2 != Y )
% 0.32/1.64 => ( ~ ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_145_sgn__eq__0__iff,axiom,
% 0.32/1.64 ! [A: real] :
% 0.32/1.64 ( ( ( sgn_sgn_real @ A )
% 0.32/1.64 = zero_zero_real )
% 0.32/1.64 = ( A = zero_zero_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_146_sgn__eq__0__iff,axiom,
% 0.32/1.64 ! [A: poly_real] :
% 0.32/1.64 ( ( ( sgn_sgn_poly_real @ A )
% 0.32/1.64 = zero_zero_poly_real )
% 0.32/1.64 = ( A = zero_zero_poly_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_147_sgn__eq__0__iff,axiom,
% 0.32/1.64 ! [A: poly_poly_real] :
% 0.32/1.64 ( ( ( sgn_sg2128174761y_real @ A )
% 0.32/1.64 = zero_z1423781445y_real )
% 0.32/1.64 = ( A = zero_z1423781445y_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_148_sgn__0__0,axiom,
% 0.32/1.64 ! [A: real] :
% 0.32/1.64 ( ( ( sgn_sgn_real @ A )
% 0.32/1.64 = zero_zero_real )
% 0.32/1.64 = ( A = zero_zero_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_149_sgn__0__0,axiom,
% 0.32/1.64 ! [A: poly_real] :
% 0.32/1.64 ( ( ( sgn_sgn_poly_real @ A )
% 0.32/1.64 = zero_zero_poly_real )
% 0.32/1.64 = ( A = zero_zero_poly_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_150_sgn__0__0,axiom,
% 0.32/1.64 ! [A: poly_poly_real] :
% 0.32/1.64 ( ( ( sgn_sg2128174761y_real @ A )
% 0.32/1.64 = zero_z1423781445y_real )
% 0.32/1.64 = ( A = zero_z1423781445y_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_151_sgn__zero__iff,axiom,
% 0.32/1.64 ! [X2: real] :
% 0.32/1.64 ( ( ( sgn_sgn_real @ X2 )
% 0.32/1.64 = zero_zero_real )
% 0.32/1.64 = ( X2 = zero_zero_real ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_152_le0,axiom,
% 0.32/1.64 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_153_bot__nat__0_Oextremum,axiom,
% 0.32/1.64 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_154_neq0__conv,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ( ( N != zero_zero_nat )
% 0.32/1.64 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_155_less__nat__zero__code,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_156_bot__nat__0_Onot__eq__extremum,axiom,
% 0.32/1.64 ! [A: nat] :
% 0.32/1.64 ( ( A != zero_zero_nat )
% 0.32/1.64 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_157_min__0R,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.32/1.64 = zero_zero_nat ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_158_min__0L,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.32/1.64 = zero_zero_nat ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_159_le__refl,axiom,
% 0.32/1.64 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_160_le__trans,axiom,
% 0.32/1.64 ! [I: nat,J: nat,K: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ I @ J )
% 0.32/1.64 => ( ( ord_less_eq_nat @ J @ K )
% 0.32/1.64 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_161_eq__imp__le,axiom,
% 0.32/1.64 ! [M: nat,N: nat] :
% 0.32/1.64 ( ( M = N )
% 0.32/1.64 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_162_le__antisym,axiom,
% 0.32/1.64 ! [M: nat,N: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ M @ N )
% 0.32/1.64 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/1.64 => ( M = N ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_163_nat__le__linear,axiom,
% 0.32/1.64 ! [M: nat,N: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ M @ N )
% 0.32/1.64 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_164_Nat_Oex__has__greatest__nat,axiom,
% 0.32/1.64 ! [P3: nat > $o,K: nat,B: nat] :
% 0.32/1.64 ( ( P3 @ K )
% 0.32/1.64 => ( ! [Y2: nat] :
% 0.32/1.64 ( ( P3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ Y2 @ B ) )
% 0.32/1.64 => ? [X3: nat] :
% 0.32/1.64 ( ( P3 @ X3 )
% 0.32/1.64 & ! [Y4: nat] :
% 0.32/1.64 ( ( P3 @ Y4 )
% 0.32/1.64 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_165_less__mono__imp__le__mono,axiom,
% 0.32/1.64 ! [F: nat > nat,I: nat,J: nat] :
% 0.32/1.64 ( ! [I2: nat,J2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ I2 @ J2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.32/1.64 => ( ( ord_less_eq_nat @ I @ J )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_166_le__neq__implies__less,axiom,
% 0.32/1.64 ! [M: nat,N: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ M @ N )
% 0.32/1.64 => ( ( M != N )
% 0.32/1.64 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_167_less__or__eq__imp__le,axiom,
% 0.32/1.64 ! [M: nat,N: nat] :
% 0.32/1.64 ( ( ( ord_less_nat @ M @ N )
% 0.32/1.64 | ( M = N ) )
% 0.32/1.64 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_168_le__eq__less__or__eq,axiom,
% 0.32/1.64 ( ord_less_eq_nat
% 0.32/1.64 = ( ^ [M2: nat,N2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ M2 @ N2 )
% 0.32/1.64 | ( M2 = N2 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_169_less__imp__le__nat,axiom,
% 0.32/1.64 ! [M: nat,N: nat] :
% 0.32/1.64 ( ( ord_less_nat @ M @ N )
% 0.32/1.64 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_170_nat__less__le,axiom,
% 0.32/1.64 ( ord_less_nat
% 0.32/1.64 = ( ^ [M2: nat,N2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.32/1.64 & ( M2 != N2 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_171_linorder__neqE__nat,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( X2 != Y )
% 0.32/1.64 => ( ~ ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_172_infinite__descent,axiom,
% 0.32/1.64 ! [P3: nat > $o,N: nat] :
% 0.32/1.64 ( ! [N3: nat] :
% 0.32/1.64 ( ~ ( P3 @ N3 )
% 0.32/1.64 => ? [M3: nat] :
% 0.32/1.64 ( ( ord_less_nat @ M3 @ N3 )
% 0.32/1.64 & ~ ( P3 @ M3 ) ) )
% 0.32/1.64 => ( P3 @ N ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_173_nat__less__induct,axiom,
% 0.32/1.64 ! [P3: nat > $o,N: nat] :
% 0.32/1.64 ( ! [N3: nat] :
% 0.32/1.64 ( ! [M3: nat] :
% 0.32/1.64 ( ( ord_less_nat @ M3 @ N3 )
% 0.32/1.64 => ( P3 @ M3 ) )
% 0.32/1.64 => ( P3 @ N3 ) )
% 0.32/1.64 => ( P3 @ N ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_174_less__irrefl__nat,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ~ ( ord_less_nat @ N @ N ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_175_less__not__refl3,axiom,
% 0.32/1.64 ! [S2: nat,T: nat] :
% 0.32/1.64 ( ( ord_less_nat @ S2 @ T )
% 0.32/1.64 => ( S2 != T ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_176_less__not__refl2,axiom,
% 0.32/1.64 ! [N: nat,M: nat] :
% 0.32/1.64 ( ( ord_less_nat @ N @ M )
% 0.32/1.64 => ( M != N ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_177_less__not__refl,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ~ ( ord_less_nat @ N @ N ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_178_nat__neq__iff,axiom,
% 0.32/1.64 ! [M: nat,N: nat] :
% 0.32/1.64 ( ( M != N )
% 0.32/1.64 = ( ( ord_less_nat @ M @ N )
% 0.32/1.64 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_179_bot__nat__0_Oextremum__strict,axiom,
% 0.32/1.64 ! [A: nat] :
% 0.32/1.64 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_180_infinite__descent0,axiom,
% 0.32/1.64 ! [P3: nat > $o,N: nat] :
% 0.32/1.64 ( ( P3 @ zero_zero_nat )
% 0.32/1.64 => ( ! [N3: nat] :
% 0.32/1.64 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.32/1.64 => ( ~ ( P3 @ N3 )
% 0.32/1.64 => ? [M3: nat] :
% 0.32/1.64 ( ( ord_less_nat @ M3 @ N3 )
% 0.32/1.64 & ~ ( P3 @ M3 ) ) ) )
% 0.32/1.64 => ( P3 @ N ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_181_gr__implies__not0,axiom,
% 0.32/1.64 ! [M: nat,N: nat] :
% 0.32/1.64 ( ( ord_less_nat @ M @ N )
% 0.32/1.64 => ( N != zero_zero_nat ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_182_less__zeroE,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_183_not__less0,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_184_not__gr0,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.32/1.64 = ( N = zero_zero_nat ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_185_gr0I,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ( ( N != zero_zero_nat )
% 0.32/1.64 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_186_less__eq__nat_Osimps_I1_J,axiom,
% 0.32/1.64 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_187_le__0__eq,axiom,
% 0.32/1.64 ! [N: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/1.64 = ( N = zero_zero_nat ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_188_ex__least__nat__le,axiom,
% 0.32/1.64 ! [P3: nat > $o,N: nat] :
% 0.32/1.64 ( ( P3 @ N )
% 0.32/1.64 => ( ~ ( P3 @ zero_zero_nat )
% 0.32/1.64 => ? [K2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ K2 @ N )
% 0.32/1.64 & ! [I3: nat] :
% 0.32/1.64 ( ( ord_less_nat @ I3 @ K2 )
% 0.32/1.64 => ~ ( P3 @ I3 ) )
% 0.32/1.64 & ( P3 @ K2 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_189_bot__nat__0_Oextremum__unique,axiom,
% 0.32/1.64 ! [A: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/1.64 = ( A = zero_zero_nat ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_190_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.32/1.64 ! [A: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/1.64 => ( A = zero_zero_nat ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_191_order__refl,axiom,
% 0.32/1.64 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_192_order__refl,axiom,
% 0.32/1.64 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_193_min__absorb1,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X2 @ Y )
% 0.32/1.64 => ( ( ord_min_real @ X2 @ Y )
% 0.32/1.64 = X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_194_min__absorb1,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.32/1.64 => ( ( ord_min_nat @ X2 @ Y )
% 0.32/1.64 = X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_195_min__absorb2,axiom,
% 0.32/1.64 ! [Y: real,X2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ Y @ X2 )
% 0.32/1.64 => ( ( ord_min_real @ X2 @ Y )
% 0.32/1.64 = Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_196_min__absorb2,axiom,
% 0.32/1.64 ! [Y: nat,X2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ Y @ X2 )
% 0.32/1.64 => ( ( ord_min_nat @ X2 @ Y )
% 0.32/1.64 = Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_197_min__def,axiom,
% 0.32/1.64 ( ord_min_real
% 0.32/1.64 = ( ^ [A2: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_198_min__def,axiom,
% 0.32/1.64 ( ord_min_nat
% 0.32/1.64 = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_199_nat__descend__induct,axiom,
% 0.32/1.64 ! [N: nat,P3: nat > $o,M: nat] :
% 0.32/1.64 ( ! [K2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ N @ K2 )
% 0.32/1.64 => ( P3 @ K2 ) )
% 0.32/1.64 => ( ! [K2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ K2 @ N )
% 0.32/1.64 => ( ! [I3: nat] :
% 0.32/1.64 ( ( ord_less_nat @ K2 @ I3 )
% 0.32/1.64 => ( P3 @ I3 ) )
% 0.32/1.64 => ( P3 @ K2 ) ) )
% 0.32/1.64 => ( P3 @ M ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_200_dual__order_Oantisym,axiom,
% 0.32/1.64 ! [B: real,A: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ B @ A )
% 0.32/1.64 => ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( A = B ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_201_dual__order_Oantisym,axiom,
% 0.32/1.64 ! [B: nat,A: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ B @ A )
% 0.32/1.64 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( A = B ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_202_dual__order_Oeq__iff,axiom,
% 0.32/1.64 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.32/1.64 = ( ^ [A2: real,B2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ B2 @ A2 )
% 0.32/1.64 & ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_203_dual__order_Oeq__iff,axiom,
% 0.32/1.64 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.32/1.64 = ( ^ [A2: nat,B2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.32/1.64 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_204_dual__order_Otrans,axiom,
% 0.32/1.64 ! [B: real,A: real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ B @ A )
% 0.32/1.64 => ( ( ord_less_eq_real @ C @ B )
% 0.32/1.64 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_205_dual__order_Otrans,axiom,
% 0.32/1.64 ! [B: nat,A: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ B @ A )
% 0.32/1.64 => ( ( ord_less_eq_nat @ C @ B )
% 0.32/1.64 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_206_linorder__wlog,axiom,
% 0.32/1.64 ! [P3: real > real > $o,A: real,B: real] :
% 0.32/1.64 ( ! [A4: real,B3: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A4 @ B3 )
% 0.32/1.64 => ( P3 @ A4 @ B3 ) )
% 0.32/1.64 => ( ! [A4: real,B3: real] :
% 0.32/1.64 ( ( P3 @ B3 @ A4 )
% 0.32/1.64 => ( P3 @ A4 @ B3 ) )
% 0.32/1.64 => ( P3 @ A @ B ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_207_linorder__wlog,axiom,
% 0.32/1.64 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 0.32/1.64 ( ! [A4: nat,B3: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A4 @ B3 )
% 0.32/1.64 => ( P3 @ A4 @ B3 ) )
% 0.32/1.64 => ( ! [A4: nat,B3: nat] :
% 0.32/1.64 ( ( P3 @ B3 @ A4 )
% 0.32/1.64 => ( P3 @ A4 @ B3 ) )
% 0.32/1.64 => ( P3 @ A @ B ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_208_dual__order_Orefl,axiom,
% 0.32/1.64 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_209_dual__order_Orefl,axiom,
% 0.32/1.64 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_210_order__trans,axiom,
% 0.32/1.64 ! [X2: real,Y: real,Z: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X2 @ Y )
% 0.32/1.64 => ( ( ord_less_eq_real @ Y @ Z )
% 0.32/1.64 => ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_211_order__trans,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat,Z: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.32/1.64 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.32/1.64 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_212_order__class_Oorder_Oantisym,axiom,
% 0.32/1.64 ! [A: real,B: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_real @ B @ A )
% 0.32/1.64 => ( A = B ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_213_order__class_Oorder_Oantisym,axiom,
% 0.32/1.64 ! [A: nat,B: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_nat @ B @ A )
% 0.32/1.64 => ( A = B ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_214_ord__le__eq__trans,axiom,
% 0.32/1.64 ! [A: real,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( B = C )
% 0.32/1.64 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_215_ord__le__eq__trans,axiom,
% 0.32/1.64 ! [A: nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( B = C )
% 0.32/1.64 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_216_ord__eq__le__trans,axiom,
% 0.32/1.64 ! [A: real,B: real,C: real] :
% 0.32/1.64 ( ( A = B )
% 0.32/1.64 => ( ( ord_less_eq_real @ B @ C )
% 0.32/1.64 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_217_ord__eq__le__trans,axiom,
% 0.32/1.64 ! [A: nat,B: nat,C: nat] :
% 0.32/1.64 ( ( A = B )
% 0.32/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/1.64 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_218_order__class_Oorder_Oeq__iff,axiom,
% 0.32/1.64 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.32/1.64 = ( ^ [A2: real,B2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A2 @ B2 )
% 0.32/1.64 & ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_219_order__class_Oorder_Oeq__iff,axiom,
% 0.32/1.64 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.32/1.64 = ( ^ [A2: nat,B2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.32/1.64 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_220_antisym__conv,axiom,
% 0.32/1.64 ! [Y: real,X2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ Y @ X2 )
% 0.32/1.64 => ( ( ord_less_eq_real @ X2 @ Y )
% 0.32/1.64 = ( X2 = Y ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_221_antisym__conv,axiom,
% 0.32/1.64 ! [Y: nat,X2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ Y @ X2 )
% 0.32/1.64 => ( ( ord_less_eq_nat @ X2 @ Y )
% 0.32/1.64 = ( X2 = Y ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_222_le__cases3,axiom,
% 0.32/1.64 ! [X2: real,Y: real,Z: real] :
% 0.32/1.64 ( ( ( ord_less_eq_real @ X2 @ Y )
% 0.32/1.64 => ~ ( ord_less_eq_real @ Y @ Z ) )
% 0.32/1.64 => ( ( ( ord_less_eq_real @ Y @ X2 )
% 0.32/1.64 => ~ ( ord_less_eq_real @ X2 @ Z ) )
% 0.32/1.64 => ( ( ( ord_less_eq_real @ X2 @ Z )
% 0.32/1.64 => ~ ( ord_less_eq_real @ Z @ Y ) )
% 0.32/1.64 => ( ( ( ord_less_eq_real @ Z @ Y )
% 0.32/1.64 => ~ ( ord_less_eq_real @ Y @ X2 ) )
% 0.32/1.64 => ( ( ( ord_less_eq_real @ Y @ Z )
% 0.32/1.64 => ~ ( ord_less_eq_real @ Z @ X2 ) )
% 0.32/1.64 => ~ ( ( ord_less_eq_real @ Z @ X2 )
% 0.32/1.64 => ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_223_le__cases3,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat,Z: nat] :
% 0.32/1.64 ( ( ( ord_less_eq_nat @ X2 @ Y )
% 0.32/1.64 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 0.32/1.64 => ( ( ( ord_less_eq_nat @ Y @ X2 )
% 0.32/1.64 => ~ ( ord_less_eq_nat @ X2 @ Z ) )
% 0.32/1.64 => ( ( ( ord_less_eq_nat @ X2 @ Z )
% 0.32/1.64 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 0.32/1.64 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 0.32/1.64 => ~ ( ord_less_eq_nat @ Y @ X2 ) )
% 0.32/1.64 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 0.32/1.64 => ~ ( ord_less_eq_nat @ Z @ X2 ) )
% 0.32/1.64 => ~ ( ( ord_less_eq_nat @ Z @ X2 )
% 0.32/1.64 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_224_order_Otrans,axiom,
% 0.32/1.64 ! [A: real,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_real @ B @ C )
% 0.32/1.64 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_225_order_Otrans,axiom,
% 0.32/1.64 ! [A: nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/1.64 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_226_le__cases,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ~ ( ord_less_eq_real @ X2 @ Y )
% 0.32/1.64 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_227_le__cases,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ~ ( ord_less_eq_nat @ X2 @ Y )
% 0.32/1.64 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_228_eq__refl,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( X2 = Y )
% 0.32/1.64 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_229_eq__refl,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( X2 = Y )
% 0.32/1.64 => ( ord_less_eq_nat @ X2 @ Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_230_linear,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X2 @ Y )
% 0.32/1.64 | ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_231_linear,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.32/1.64 | ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_232_antisym,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X2 @ Y )
% 0.32/1.64 => ( ( ord_less_eq_real @ Y @ X2 )
% 0.32/1.64 => ( X2 = Y ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_233_antisym,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.32/1.64 => ( ( ord_less_eq_nat @ Y @ X2 )
% 0.32/1.64 => ( X2 = Y ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_234_eq__iff,axiom,
% 0.32/1.64 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.32/1.64 = ( ^ [X4: real,Y3: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.32/1.64 & ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_235_eq__iff,axiom,
% 0.32/1.64 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.32/1.64 = ( ^ [X4: nat,Y3: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.32/1.64 & ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_236_ord__le__eq__subst,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( ( F @ B )
% 0.32/1.64 = C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_237_ord__le__eq__subst,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( ( F @ B )
% 0.32/1.64 = C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_238_ord__le__eq__subst,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( ( F @ B )
% 0.32/1.64 = C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_239_ord__le__eq__subst,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( ( F @ B )
% 0.32/1.64 = C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_240_ord__eq__le__subst,axiom,
% 0.32/1.64 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/1.64 ( ( A
% 0.32/1.64 = ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_241_ord__eq__le__subst,axiom,
% 0.32/1.64 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.32/1.64 ( ( A
% 0.32/1.64 = ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_242_ord__eq__le__subst,axiom,
% 0.32/1.64 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/1.64 ( ( A
% 0.32/1.64 = ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_243_ord__eq__le__subst,axiom,
% 0.32/1.64 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.32/1.64 ( ( A
% 0.32/1.64 = ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_244_order__subst2,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_245_order__subst2,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_246_order__subst2,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_247_order__subst2,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_248_order__subst1,axiom,
% 0.32/1.64 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_249_order__subst1,axiom,
% 0.32/1.64 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_250_order__subst1,axiom,
% 0.32/1.64 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_251_order__subst1,axiom,
% 0.32/1.64 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_252_ord__eq__less__subst,axiom,
% 0.32/1.64 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/1.64 ( ( A
% 0.32/1.64 = ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_253_ord__eq__less__subst,axiom,
% 0.32/1.64 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.32/1.64 ( ( A
% 0.32/1.64 = ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_254_ord__eq__less__subst,axiom,
% 0.32/1.64 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/1.64 ( ( A
% 0.32/1.64 = ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_255_ord__eq__less__subst,axiom,
% 0.32/1.64 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.32/1.64 ( ( A
% 0.32/1.64 = ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_256_ord__less__eq__subst,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/1.64 ( ( ord_less_real @ A @ B )
% 0.32/1.64 => ( ( ( F @ B )
% 0.32/1.64 = C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_257_ord__less__eq__subst,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_real @ A @ B )
% 0.32/1.64 => ( ( ( F @ B )
% 0.32/1.64 = C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_258_ord__less__eq__subst,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.32/1.64 ( ( ord_less_nat @ A @ B )
% 0.32/1.64 => ( ( ( F @ B )
% 0.32/1.64 = C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_259_ord__less__eq__subst,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A @ B )
% 0.32/1.64 => ( ( ( F @ B )
% 0.32/1.64 = C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_260_order__less__subst1,axiom,
% 0.32/1.64 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_261_order__less__subst1,axiom,
% 0.32/1.64 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_262_order__less__subst1,axiom,
% 0.32/1.64 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_263_order__less__subst1,axiom,
% 0.32/1.64 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_264_order__less__subst2,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/1.64 ( ( ord_less_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_265_order__less__subst2,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_266_order__less__subst2,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.32/1.64 ( ( ord_less_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_267_order__less__subst2,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_268_lt__ex,axiom,
% 0.32/1.64 ! [X2: real] :
% 0.32/1.64 ? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_269_gt__ex,axiom,
% 0.32/1.64 ! [X2: real] :
% 0.32/1.64 ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_270_gt__ex,axiom,
% 0.32/1.64 ! [X2: nat] :
% 0.32/1.64 ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_271_neqE,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( X2 != Y )
% 0.32/1.64 => ( ~ ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_272_neqE,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( X2 != Y )
% 0.32/1.64 => ( ~ ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_273_neq__iff,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( X2 != Y )
% 0.32/1.64 = ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 | ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_274_neq__iff,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( X2 != Y )
% 0.32/1.64 = ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 | ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_275_order_Oasym,axiom,
% 0.32/1.64 ! [A: real,B: real] :
% 0.32/1.64 ( ( ord_less_real @ A @ B )
% 0.32/1.64 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_276_order_Oasym,axiom,
% 0.32/1.64 ! [A: nat,B: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A @ B )
% 0.32/1.64 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_277_dense,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ? [Z5: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Z5 )
% 0.32/1.64 & ( ord_less_real @ Z5 @ Y ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_278_less__imp__neq,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ( X2 != Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_279_less__imp__neq,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ( X2 != Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_280_less__asym,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_281_less__asym,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_282_less__asym_H,axiom,
% 0.32/1.64 ! [A: real,B: real] :
% 0.32/1.64 ( ( ord_less_real @ A @ B )
% 0.32/1.64 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_283_less__asym_H,axiom,
% 0.32/1.64 ! [A: nat,B: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A @ B )
% 0.32/1.64 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_284_less__trans,axiom,
% 0.32/1.64 ! [X2: real,Y: real,Z: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ( ( ord_less_real @ Y @ Z )
% 0.32/1.64 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_285_less__trans,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat,Z: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ( ( ord_less_nat @ Y @ Z )
% 0.32/1.64 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_286_less__linear,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 | ( X2 = Y )
% 0.32/1.64 | ( ord_less_real @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_287_less__linear,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 | ( X2 = Y )
% 0.32/1.64 | ( ord_less_nat @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_288_less__irrefl,axiom,
% 0.32/1.64 ! [X2: real] :
% 0.32/1.64 ~ ( ord_less_real @ X2 @ X2 ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_289_less__irrefl,axiom,
% 0.32/1.64 ! [X2: nat] :
% 0.32/1.64 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_290_ord__eq__less__trans,axiom,
% 0.32/1.64 ! [A: real,B: real,C: real] :
% 0.32/1.64 ( ( A = B )
% 0.32/1.64 => ( ( ord_less_real @ B @ C )
% 0.32/1.64 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_291_ord__eq__less__trans,axiom,
% 0.32/1.64 ! [A: nat,B: nat,C: nat] :
% 0.32/1.64 ( ( A = B )
% 0.32/1.64 => ( ( ord_less_nat @ B @ C )
% 0.32/1.64 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_292_ord__less__eq__trans,axiom,
% 0.32/1.64 ! [A: real,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_real @ A @ B )
% 0.32/1.64 => ( ( B = C )
% 0.32/1.64 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_293_ord__less__eq__trans,axiom,
% 0.32/1.64 ! [A: nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A @ B )
% 0.32/1.64 => ( ( B = C )
% 0.32/1.64 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_294_dual__order_Oasym,axiom,
% 0.32/1.64 ! [B: real,A: real] :
% 0.32/1.64 ( ( ord_less_real @ B @ A )
% 0.32/1.64 => ~ ( ord_less_real @ A @ B ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_295_dual__order_Oasym,axiom,
% 0.32/1.64 ! [B: nat,A: nat] :
% 0.32/1.64 ( ( ord_less_nat @ B @ A )
% 0.32/1.64 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_296_less__imp__not__eq,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ( X2 != Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_297_less__imp__not__eq,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ( X2 != Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_298_less__not__sym,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_299_less__not__sym,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_300_less__induct,axiom,
% 0.32/1.64 ! [P3: nat > $o,A: nat] :
% 0.32/1.64 ( ! [X3: nat] :
% 0.32/1.64 ( ! [Y4: nat] :
% 0.32/1.64 ( ( ord_less_nat @ Y4 @ X3 )
% 0.32/1.64 => ( P3 @ Y4 ) )
% 0.32/1.64 => ( P3 @ X3 ) )
% 0.32/1.64 => ( P3 @ A ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_301_antisym__conv3,axiom,
% 0.32/1.64 ! [Y: real,X2: real] :
% 0.32/1.64 ( ~ ( ord_less_real @ Y @ X2 )
% 0.32/1.64 => ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 0.32/1.64 = ( X2 = Y ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_302_antisym__conv3,axiom,
% 0.32/1.64 ! [Y: nat,X2: nat] :
% 0.32/1.64 ( ~ ( ord_less_nat @ Y @ X2 )
% 0.32/1.64 => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 0.32/1.64 = ( X2 = Y ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_303_less__imp__not__eq2,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ( Y != X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_304_less__imp__not__eq2,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ( Y != X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_305_less__imp__triv,axiom,
% 0.32/1.64 ! [X2: real,Y: real,P3: $o] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ( ( ord_less_real @ Y @ X2 )
% 0.32/1.64 => P3 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_306_less__imp__triv,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat,P3: $o] :
% 0.32/1.64 ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ( ( ord_less_nat @ Y @ X2 )
% 0.32/1.64 => P3 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_307_linorder__cases,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ~ ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ( ( X2 != Y )
% 0.32/1.64 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_308_linorder__cases,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ~ ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ( ( X2 != Y )
% 0.32/1.64 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_309_dual__order_Oirrefl,axiom,
% 0.32/1.64 ! [A: real] :
% 0.32/1.64 ~ ( ord_less_real @ A @ A ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_310_dual__order_Oirrefl,axiom,
% 0.32/1.64 ! [A: nat] :
% 0.32/1.64 ~ ( ord_less_nat @ A @ A ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_311_order_Ostrict__trans,axiom,
% 0.32/1.64 ! [A: real,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_real @ B @ C )
% 0.32/1.64 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_312_order_Ostrict__trans,axiom,
% 0.32/1.64 ! [A: nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_nat @ B @ C )
% 0.32/1.64 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_313_less__imp__not__less,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_314_less__imp__not__less,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_315_exists__least__iff,axiom,
% 0.32/1.64 ( ( ^ [P4: nat > $o] :
% 0.32/1.64 ? [X5: nat] : ( P4 @ X5 ) )
% 0.32/1.64 = ( ^ [P5: nat > $o] :
% 0.32/1.64 ? [N2: nat] :
% 0.32/1.64 ( ( P5 @ N2 )
% 0.32/1.64 & ! [M2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ M2 @ N2 )
% 0.32/1.64 => ~ ( P5 @ M2 ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_316_linorder__less__wlog,axiom,
% 0.32/1.64 ! [P3: real > real > $o,A: real,B: real] :
% 0.32/1.64 ( ! [A4: real,B3: real] :
% 0.32/1.64 ( ( ord_less_real @ A4 @ B3 )
% 0.32/1.64 => ( P3 @ A4 @ B3 ) )
% 0.32/1.64 => ( ! [A4: real] : ( P3 @ A4 @ A4 )
% 0.32/1.64 => ( ! [A4: real,B3: real] :
% 0.32/1.64 ( ( P3 @ B3 @ A4 )
% 0.32/1.64 => ( P3 @ A4 @ B3 ) )
% 0.32/1.64 => ( P3 @ A @ B ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_317_linorder__less__wlog,axiom,
% 0.32/1.64 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 0.32/1.64 ( ! [A4: nat,B3: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A4 @ B3 )
% 0.32/1.64 => ( P3 @ A4 @ B3 ) )
% 0.32/1.64 => ( ! [A4: nat] : ( P3 @ A4 @ A4 )
% 0.32/1.64 => ( ! [A4: nat,B3: nat] :
% 0.32/1.64 ( ( P3 @ B3 @ A4 )
% 0.32/1.64 => ( P3 @ A4 @ B3 ) )
% 0.32/1.64 => ( P3 @ A @ B ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_318_dual__order_Ostrict__trans,axiom,
% 0.32/1.64 ! [B: real,A: real,C: real] :
% 0.32/1.64 ( ( ord_less_real @ B @ A )
% 0.32/1.64 => ( ( ord_less_real @ C @ B )
% 0.32/1.64 => ( ord_less_real @ C @ A ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_319_dual__order_Ostrict__trans,axiom,
% 0.32/1.64 ! [B: nat,A: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_nat @ B @ A )
% 0.32/1.64 => ( ( ord_less_nat @ C @ B )
% 0.32/1.64 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_320_not__less__iff__gr__or__eq,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 0.32/1.64 = ( ( ord_less_real @ Y @ X2 )
% 0.32/1.64 | ( X2 = Y ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_321_not__less__iff__gr__or__eq,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 0.32/1.64 = ( ( ord_less_nat @ Y @ X2 )
% 0.32/1.64 | ( X2 = Y ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_322_order_Ostrict__implies__not__eq,axiom,
% 0.32/1.64 ! [A: real,B: real] :
% 0.32/1.64 ( ( ord_less_real @ A @ B )
% 0.32/1.64 => ( A != B ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_323_order_Ostrict__implies__not__eq,axiom,
% 0.32/1.64 ! [A: nat,B: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A @ B )
% 0.32/1.64 => ( A != B ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_324_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/1.64 ! [B: real,A: real] :
% 0.32/1.64 ( ( ord_less_real @ B @ A )
% 0.32/1.64 => ( A != B ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_325_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/1.64 ! [B: nat,A: nat] :
% 0.32/1.64 ( ( ord_less_nat @ B @ A )
% 0.32/1.64 => ( A != B ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_326_leD,axiom,
% 0.32/1.64 ! [Y: real,X2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ Y @ X2 )
% 0.32/1.64 => ~ ( ord_less_real @ X2 @ Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_327_leD,axiom,
% 0.32/1.64 ! [Y: nat,X2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ Y @ X2 )
% 0.32/1.64 => ~ ( ord_less_nat @ X2 @ Y ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_328_leI,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ~ ( ord_less_real @ X2 @ Y )
% 0.32/1.64 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_329_leI,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ~ ( ord_less_nat @ X2 @ Y )
% 0.32/1.64 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_330_le__less,axiom,
% 0.32/1.64 ( ord_less_eq_real
% 0.32/1.64 = ( ^ [X4: real,Y3: real] :
% 0.32/1.64 ( ( ord_less_real @ X4 @ Y3 )
% 0.32/1.64 | ( X4 = Y3 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_331_le__less,axiom,
% 0.32/1.64 ( ord_less_eq_nat
% 0.32/1.64 = ( ^ [X4: nat,Y3: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X4 @ Y3 )
% 0.32/1.64 | ( X4 = Y3 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_332_less__le,axiom,
% 0.32/1.64 ( ord_less_real
% 0.32/1.64 = ( ^ [X4: real,Y3: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.32/1.64 & ( X4 != Y3 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_333_less__le,axiom,
% 0.32/1.64 ( ord_less_nat
% 0.32/1.64 = ( ^ [X4: nat,Y3: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.32/1.64 & ( X4 != Y3 ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_334_order__le__less__subst1,axiom,
% 0.32/1.64 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_335_order__le__less__subst1,axiom,
% 0.32/1.64 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_336_order__le__less__subst1,axiom,
% 0.32/1.64 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_337_order__le__less__subst1,axiom,
% 0.32/1.64 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_338_order__le__less__subst2,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_339_order__le__less__subst2,axiom,
% 0.32/1.64 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.64 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_340_order__le__less__subst2,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_341_order__le__less__subst2,axiom,
% 0.32/1.64 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ A @ B )
% 0.32/1.64 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_342_order__less__le__subst1,axiom,
% 0.32/1.64 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.32/1.64 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_real @ B @ C )
% 0.32/1.64 => ( ! [X3: real,Y2: real] :
% 0.32/1.64 ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_343_order__less__le__subst1,axiom,
% 0.32/1.64 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(fact_344_order__less__le__subst1,axiom,
% 0.32/1.64 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.32/1.64 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.32/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/1.64 => ( ! [X3: nat,Y2: nat] :
% 0.32/1.64 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.32/1.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/1.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( if_nat @ $false @ X2 @ Y )
% 0.32/1.64 = Y ) ).
% 0.32/1.64
% 0.32/1.64 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.32/1.64 ! [X2: nat,Y: nat] :
% 0.32/1.64 ( ( if_nat @ $true @ X2 @ Y )
% 0.32/1.64 = X2 ) ).
% 0.32/1.64
% 0.32/1.64 thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
% 0.32/1.64 ! [P3: $o] :
% 0.32/1.64 ( ( P3 = $true )
% 0.32/1.64 | ( P3 = $false ) ) ).
% 0.32/1.64
% 0.32/1.64 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( if_real @ $false @ X2 @ Y )
% 0.32/1.64 = Y ) ).
% 0.32/1.64
% 0.32/1.64 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.32/1.64 ! [X2: real,Y: real] :
% 0.32/1.64 ( ( if_real @ $true @ X2 @ Y )
% 0.32/1.64 = X2 ) ).
% 0.32/1.64
% 0.32/1.64 thf(conj_0,conjecture,
% 0.32/1.64 ! [X3: real] :
% 0.32/1.64 ( ( ( poly_real2 @ p @ X3 )
% 0.32/1.64 != zero_zero_real )
% 0.32/1.64 | ( ord_less_real @ lb @ X3 ) ) ).
% 0.32/1.64 ------------------------
% 0.32/1.66 % (18531)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/183Mi)
% 0.32/1.67 % (18535)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_18417 for (2999ds/2Mi)
% 0.32/1.67 % (18532)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_18417 for (2999ds/4Mi)
% 0.32/1.67 % (18533)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_18417 for (2999ds/27Mi)
% 0.32/1.67 % (18535)Instruction limit reached!
% 0.32/1.67 % (18535)------------------------------
% 0.32/1.67 % (18535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.67 % (18535)Termination reason: Unknown
% 0.32/1.67 % (18535)Termination phase: shuffling
% 0.32/1.67
% 0.32/1.67 % (18535)Memory used [KB]: 1407
% 0.32/1.67 % (18535)Time elapsed: 0.002 s
% 0.32/1.67 % (18535)Instructions burned: 3 (million)
% 0.32/1.67 % (18535)------------------------------
% 0.32/1.67 % (18535)------------------------------
% 0.32/1.67 % (18532)Instruction limit reached!
% 0.32/1.67 % (18532)------------------------------
% 0.32/1.67 % (18532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.67 % (18532)Termination reason: Unknown
% 0.32/1.67 % (18532)Termination phase: shuffling
% 0.32/1.67
% 0.32/1.67 % (18532)Memory used [KB]: 1407
% 0.32/1.67 % (18532)Time elapsed: 0.003 s
% 0.32/1.67 % (18532)Instructions burned: 7 (million)
% 0.32/1.67 % (18532)------------------------------
% 0.32/1.67 % (18532)------------------------------
% 0.32/1.67 % (18534)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/2Mi)
% 0.32/1.67 % (18536)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_18417 for (2999ds/275Mi)
% 0.32/1.67 % (18537)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/18Mi)
% 0.32/1.67 % (18533)Instruction limit reached!
% 0.32/1.67 % (18533)------------------------------
% 0.32/1.67 % (18533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.67 % (18533)Termination reason: Unknown
% 0.32/1.67 % (18533)Termination phase: Property scanning
% 0.32/1.67
% 0.32/1.67 % (18533)Memory used [KB]: 1791
% 0.32/1.67 % (18533)Time elapsed: 0.010 s
% 0.32/1.67 % (18533)Instructions burned: 30 (million)
% 0.32/1.67 % (18533)------------------------------
% 0.32/1.67 % (18533)------------------------------
% 0.32/1.67 % (18534)Instruction limit reached!
% 0.32/1.67 % (18534)------------------------------
% 0.32/1.67 % (18534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.67 % (18534)Termination reason: Unknown
% 0.32/1.67 % (18534)Termination phase: shuffling
% 0.32/1.67
% 0.32/1.67 % (18534)Memory used [KB]: 1407
% 0.32/1.67 % (18534)Time elapsed: 0.004 s
% 0.32/1.67 % (18534)Instructions burned: 3 (million)
% 0.32/1.67 % (18534)------------------------------
% 0.32/1.67 % (18534)------------------------------
% 0.32/1.68 % (18538)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/3Mi)
% 0.32/1.68 % (18538)Instruction limit reached!
% 0.32/1.68 % (18538)------------------------------
% 0.32/1.68 % (18538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.68 % (18538)Termination reason: Unknown
% 0.32/1.68 % (18538)Termination phase: shuffling
% 0.32/1.68
% 0.32/1.68 % (18538)Memory used [KB]: 1407
% 0.32/1.68 % (18538)Time elapsed: 0.002 s
% 0.32/1.68 % (18538)Instructions burned: 3 (million)
% 0.32/1.68 % (18538)------------------------------
% 0.32/1.68 % (18538)------------------------------
% 0.32/1.68 % (18539)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_18417 for (2999ds/37Mi)
% 0.32/1.68 % (18540)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_18417 for (2999ds/15Mi)
% 0.32/1.68 % (18537)Instruction limit reached!
% 0.32/1.68 % (18537)------------------------------
% 0.32/1.68 % (18537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.68 % (18537)Termination reason: Unknown
% 0.32/1.68 % (18537)Termination phase: shuffling
% 0.32/1.68
% 0.32/1.68 % (18537)Memory used [KB]: 1663
% 0.32/1.68 % (18537)Time elapsed: 0.013 s
% 0.32/1.68 % (18537)Instructions burned: 18 (million)
% 0.32/1.68 % (18537)------------------------------
% 0.32/1.68 % (18537)------------------------------
% 0.32/1.69 % (18542)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_18417 for (2999ds/1041Mi)
% 0.32/1.69 % (18539)Instruction limit reached!
% 0.32/1.69 % (18539)------------------------------
% 0.32/1.69 % (18539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.69 % (18539)Termination reason: Unknown
% 0.32/1.69 % (18539)Termination phase: shuffling
% 0.32/1.69
% 0.32/1.69 % (18539)Memory used [KB]: 2174
% 0.32/1.69 % (18539)Time elapsed: 0.013 s
% 0.32/1.69 % (18539)Instructions burned: 39 (million)
% 0.32/1.69 % (18539)------------------------------
% 0.32/1.69 % (18539)------------------------------
% 0.32/1.69 % (18540)Instruction limit reached!
% 0.32/1.69 % (18540)------------------------------
% 0.32/1.69 % (18540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.69 % (18540)Termination reason: Unknown
% 0.32/1.69 % (18540)Termination phase: shuffling
% 0.32/1.69
% 0.32/1.69 % (18540)Memory used [KB]: 1663
% 0.32/1.69 % (18540)Time elapsed: 0.006 s
% 0.32/1.69 % (18540)Instructions burned: 16 (million)
% 0.32/1.69 % (18540)------------------------------
% 0.32/1.69 % (18540)------------------------------
% 0.32/1.69 % (18541)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_18417 for (2999ds/3Mi)
% 0.32/1.69 % (18541)Instruction limit reached!
% 0.32/1.69 % (18541)------------------------------
% 0.32/1.69 % (18541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.69 % (18541)Termination reason: Unknown
% 0.32/1.69 % (18541)Termination phase: shuffling
% 0.32/1.69
% 0.32/1.69 % (18541)Memory used [KB]: 1407
% 0.32/1.69 % (18541)Time elapsed: 0.003 s
% 0.32/1.69 % (18541)Instructions burned: 4 (million)
% 0.32/1.69 % (18541)------------------------------
% 0.32/1.69 % (18541)------------------------------
% 3.13/1.70 % (18545)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/3Mi)
% 3.13/1.70 % (18544)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/16Mi)
% 3.13/1.70 % (18545)Instruction limit reached!
% 3.13/1.70 % (18545)------------------------------
% 3.13/1.70 % (18545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.70 % (18545)Termination reason: Unknown
% 3.13/1.70 % (18545)Termination phase: shuffling
% 3.13/1.70
% 3.13/1.70 % (18545)Memory used [KB]: 1407
% 3.13/1.70 % (18545)Time elapsed: 0.002 s
% 3.13/1.70 % (18545)Instructions burned: 3 (million)
% 3.13/1.70 % (18545)------------------------------
% 3.13/1.70 % (18545)------------------------------
% 3.13/1.70 % (18543)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/7Mi)
% 3.13/1.70 % (18546)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/3Mi)
% 3.13/1.70 % (18543)Instruction limit reached!
% 3.13/1.70 % (18543)------------------------------
% 3.13/1.70 % (18543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.70 % (18543)Termination reason: Unknown
% 3.13/1.70 % (18543)Termination phase: shuffling
% 3.13/1.70
% 3.13/1.70 % (18543)Memory used [KB]: 1407
% 3.13/1.70 % (18543)Time elapsed: 0.005 s
% 3.13/1.70 % (18543)Instructions burned: 7 (million)
% 3.13/1.70 % (18543)------------------------------
% 3.13/1.70 % (18543)------------------------------
% 3.13/1.70 % (18546)Instruction limit reached!
% 3.13/1.70 % (18546)------------------------------
% 3.13/1.70 % (18546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.70 % (18546)Termination reason: Unknown
% 3.13/1.70 % (18546)Termination phase: shuffling
% 3.13/1.70
% 3.13/1.70 % (18546)Memory used [KB]: 1407
% 3.13/1.70 % (18546)Time elapsed: 0.003 s
% 3.13/1.70 % (18546)Instructions burned: 4 (million)
% 3.13/1.70 % (18546)------------------------------
% 3.13/1.70 % (18546)------------------------------
% 3.13/1.71 % (18544)Instruction limit reached!
% 3.13/1.71 % (18544)------------------------------
% 3.13/1.71 % (18544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.71 % (18544)Termination reason: Unknown
% 3.13/1.71 % (18544)Termination phase: shuffling
% 3.13/1.71
% 3.13/1.71 % (18544)Memory used [KB]: 1663
% 3.13/1.71 % (18544)Time elapsed: 0.008 s
% 3.13/1.71 % (18544)Instructions burned: 18 (million)
% 3.13/1.71 % (18544)------------------------------
% 3.13/1.71 % (18544)------------------------------
% 3.13/1.71 % (18547)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/7Mi)
% 3.13/1.71 % (18547)Instruction limit reached!
% 3.13/1.71 % (18547)------------------------------
% 3.13/1.71 % (18547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.71 % (18547)Termination reason: Unknown
% 3.13/1.71 % (18547)Termination phase: shuffling
% 3.13/1.71
% 3.13/1.71 % (18547)Memory used [KB]: 1407
% 3.13/1.71 % (18547)Time elapsed: 0.003 s
% 3.13/1.71 % (18547)Instructions burned: 7 (million)
% 3.13/1.71 % (18547)------------------------------
% 3.13/1.71 % (18547)------------------------------
% 3.13/1.71 % (18531)Instruction limit reached!
% 3.13/1.71 % (18531)------------------------------
% 3.13/1.71 % (18531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.71 % (18531)Termination reason: Unknown
% 3.13/1.71 % (18531)Termination phase: Saturation
% 3.13/1.71
% 3.13/1.71 % (18531)Memory used [KB]: 7291
% 3.13/1.71 % (18531)Time elapsed: 0.050 s
% 3.13/1.71 % (18531)Instructions burned: 184 (million)
% 3.13/1.71 % (18531)------------------------------
% 3.13/1.71 % (18531)------------------------------
% 3.13/1.71 % (18549)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/4Mi)
% 3.13/1.72 % (18548)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/3Mi)
% 3.13/1.72 % (18550)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/18Mi)
% 3.13/1.72 % (18549)Instruction limit reached!
% 3.13/1.72 % (18549)------------------------------
% 3.13/1.72 % (18549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.72 % (18549)Termination reason: Unknown
% 3.13/1.72 % (18549)Termination phase: shuffling
% 3.13/1.72
% 3.13/1.72 % (18549)Memory used [KB]: 1407
% 3.13/1.72 % (18549)Time elapsed: 0.003 s
% 3.13/1.72 % (18549)Instructions burned: 6 (million)
% 3.13/1.72 % (18549)------------------------------
% 3.13/1.72 % (18549)------------------------------
% 3.13/1.72 % (18548)Instruction limit reached!
% 3.13/1.72 % (18548)------------------------------
% 3.13/1.72 % (18548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.72 % (18548)Termination reason: Unknown
% 3.13/1.72 % (18548)Termination phase: shuffling
% 3.13/1.72
% 3.13/1.72 % (18548)Memory used [KB]: 1407
% 3.13/1.72 % (18548)Time elapsed: 0.002 s
% 3.13/1.72 % (18548)Instructions burned: 3 (million)
% 3.13/1.72 % (18548)------------------------------
% 3.13/1.72 % (18548)------------------------------
% 3.13/1.72 % (18551)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/710Mi)
% 3.13/1.72 % (18550)Instruction limit reached!
% 3.13/1.72 % (18550)------------------------------
% 3.13/1.72 % (18550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.72 % (18550)Termination reason: Unknown
% 3.13/1.72 % (18550)Termination phase: shuffling
% 3.13/1.72
% 3.13/1.72 % (18550)Memory used [KB]: 1663
% 3.13/1.72 % (18550)Time elapsed: 0.008 s
% 3.13/1.72 % (18550)Instructions burned: 18 (million)
% 3.13/1.72 % (18550)------------------------------
% 3.13/1.72 % (18550)------------------------------
% 3.13/1.72 % (18552)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/6Mi)
% 3.13/1.73 % (18552)Instruction limit reached!
% 3.13/1.73 % (18552)------------------------------
% 3.13/1.73 % (18552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.73 % (18552)Termination reason: Unknown
% 3.13/1.73 % (18552)Termination phase: shuffling
% 3.13/1.73
% 3.13/1.73 % (18552)Memory used [KB]: 1407
% 3.13/1.73 % (18552)Time elapsed: 0.003 s
% 3.13/1.73 % (18552)Instructions burned: 7 (million)
% 3.13/1.73 % (18552)------------------------------
% 3.13/1.73 % (18552)------------------------------
% 3.13/1.73 % (18553)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/902Mi)
% 3.13/1.73 % (18554)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/21Mi)
% 3.13/1.73 % (18555)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/5Mi)
% 3.13/1.73 % (18554)Instruction limit reached!
% 3.13/1.73 % (18554)------------------------------
% 3.13/1.73 % (18554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.73 % (18554)Termination reason: Unknown
% 3.13/1.73 % (18554)Termination phase: shuffling
% 3.13/1.73
% 3.13/1.73 % (18554)Memory used [KB]: 1663
% 3.13/1.73 % (18554)Time elapsed: 0.009 s
% 3.13/1.73 % (18554)Instructions burned: 21 (million)
% 3.13/1.73 % (18554)------------------------------
% 3.13/1.73 % (18554)------------------------------
% 3.13/1.73 % (18555)Instruction limit reached!
% 3.13/1.73 % (18555)------------------------------
% 3.13/1.73 % (18555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.73 % (18555)Termination reason: Unknown
% 3.13/1.73 % (18555)Termination phase: shuffling
% 3.13/1.73
% 3.13/1.73 % (18555)Memory used [KB]: 1407
% 3.13/1.73 % (18555)Time elapsed: 0.004 s
% 3.13/1.73 % (18555)Instructions burned: 7 (million)
% 3.13/1.73 % (18555)------------------------------
% 3.13/1.73 % (18555)------------------------------
% 3.13/1.73 % (18556)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_18417 for (2999ds/6Mi)
% 3.13/1.74 % (18556)Instruction limit reached!
% 3.13/1.74 % (18556)------------------------------
% 3.13/1.74 % (18556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.74 % (18556)Termination reason: Unknown
% 3.13/1.74 % (18556)Termination phase: shuffling
% 3.13/1.74
% 3.13/1.74 % (18556)Memory used [KB]: 1407
% 3.13/1.74 % (18556)Time elapsed: 0.003 s
% 3.13/1.74 % (18556)Instructions burned: 7 (million)
% 3.13/1.74 % (18556)------------------------------
% 3.13/1.74 % (18556)------------------------------
% 3.13/1.74 % (18536)First to succeed.
% 3.13/1.74 % (18536)Refutation found. Thanks to Tanya!
% 3.13/1.74 % SZS status Theorem for DTF2THF_18417
% 3.13/1.74 % SZS output start Proof for DTF2THF_18417
% 3.13/1.74 thf(type_def_5, type, poly_poly_real: $tType).
% 3.13/1.74 thf(type_def_7, type, poly_real: $tType).
% 3.13/1.74 thf(type_def_8, type, real: $tType).
% 3.13/1.74 thf(type_def_9, type, nat: $tType).
% 3.13/1.74 thf(type_def_10, type, poly_nat: $tType).
% 3.13/1.74 thf(type_def_11, type, poly_poly_nat: $tType).
% 3.13/1.74 thf(type_def_12, type, poly_poly_poly_real: $tType).
% 3.13/1.74 thf(type_def_13, type, set_real: $tType).
% 3.13/1.74 thf(func_def_0, type, poly_poly_poly_real: $tType).
% 3.13/1.74 thf(func_def_1, type, poly_poly_real: $tType).
% 3.13/1.74 thf(func_def_2, type, poly_poly_nat: $tType).
% 3.13/1.74 thf(func_def_3, type, poly_real: $tType).
% 3.13/1.74 thf(func_def_4, type, poly_nat: $tType).
% 3.13/1.74 thf(func_def_5, type, set_real: $tType).
% 3.13/1.74 thf(func_def_6, type, real: $tType).
% 3.13/1.74 thf(func_def_7, type, nat: $tType).
% 3.13/1.74 thf(func_def_8, type, sgn_sg2128174761y_real: poly_poly_real > poly_poly_real).
% 3.13/1.74 thf(func_def_9, type, sgn_sgn_poly_real: poly_real > poly_real).
% 3.13/1.74 thf(func_def_10, type, sgn_sgn_real: real > real).
% 3.13/1.74 thf(func_def_11, type, zero_zero_nat: nat).
% 3.13/1.74 thf(func_def_12, type, zero_zero_poly_nat: poly_nat).
% 3.13/1.74 thf(func_def_13, type, zero_z1059985641ly_nat: poly_poly_nat).
% 3.13/1.74 thf(func_def_14, type, zero_z935034829y_real: poly_poly_poly_real).
% 3.13/1.74 thf(func_def_15, type, zero_z1423781445y_real: poly_poly_real).
% 3.13/1.74 thf(func_def_16, type, zero_zero_poly_real: poly_real).
% 3.13/1.74 thf(func_def_17, type, zero_zero_real: real).
% 3.13/1.74 thf(func_def_18, type, if_nat: $o > nat > nat > nat).
% 3.13/1.74 thf(func_def_19, type, if_real: $o > real > real > real).
% 3.13/1.74 thf(func_def_20, type, ord_less_nat: nat > nat > $o).
% 3.13/1.74 thf(func_def_21, type, ord_le38482960y_real: poly_poly_real > poly_poly_real > $o).
% 3.13/1.74 thf(func_def_22, type, ord_less_poly_real: poly_real > poly_real > $o).
% 3.13/1.74 thf(func_def_23, type, ord_less_real: real > real > $o).
% 3.13/1.74 thf(func_def_24, type, ord_less_eq_nat: nat > nat > $o).
% 3.13/1.74 thf(func_def_25, type, ord_le893774876y_real: poly_poly_real > poly_poly_real > $o).
% 3.13/1.74 thf(func_def_26, type, ord_le1180086932y_real: poly_real > poly_real > $o).
% 3.13/1.74 thf(func_def_27, type, ord_less_eq_real: real > real > $o).
% 3.13/1.74 thf(func_def_28, type, ord_min_nat: nat > nat > nat).
% 3.13/1.74 thf(func_def_29, type, ord_min_real: real > real > real).
% 3.13/1.74 thf(func_def_30, type, divide924636027y_real: poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > nat > nat > poly_poly_poly_real).
% 3.13/1.74 thf(func_def_31, type, divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real).
% 3.13/1.74 thf(func_def_32, type, divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real).
% 3.13/1.74 thf(func_def_33, type, is_zero_nat: poly_nat > $o).
% 3.13/1.74 thf(func_def_34, type, is_zero_poly_real: poly_poly_real > $o).
% 3.13/1.74 thf(func_def_35, type, is_zero_real: poly_real > $o).
% 3.13/1.74 thf(func_def_36, type, order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat).
% 3.13/1.74 thf(func_def_37, type, order_poly_real: poly_real > poly_poly_real > nat).
% 3.13/1.74 thf(func_def_38, type, order_real: real > poly_real > nat).
% 3.13/1.74 thf(func_def_39, type, poly_nat2: poly_nat > nat > nat).
% 3.13/1.74 thf(func_def_40, type, poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat).
% 3.13/1.74 thf(func_def_41, type, poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real).
% 3.13/1.74 thf(func_def_42, type, poly_poly_real2: poly_poly_real > poly_real > poly_real).
% 3.13/1.74 thf(func_def_43, type, poly_real2: poly_real > real > real).
% 3.13/1.74 thf(func_def_44, type, poly_cutoff_nat: nat > poly_nat > poly_nat).
% 3.13/1.74 thf(func_def_45, type, poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real).
% 3.13/1.74 thf(func_def_46, type, poly_cutoff_real: nat > poly_real > poly_real).
% 3.13/1.74 thf(func_def_47, type, poly_shift_nat: nat > poly_nat > poly_nat).
% 3.13/1.74 thf(func_def_48, type, poly_shift_poly_real: nat > poly_poly_real > poly_poly_real).
% 3.13/1.74 thf(func_def_49, type, poly_shift_real: nat > poly_real > poly_real).
% 3.13/1.74 thf(func_def_50, type, reflect_poly_nat: poly_nat > poly_nat).
% 3.13/1.74 thf(func_def_51, type, reflec781175074ly_nat: poly_poly_nat > poly_poly_nat).
% 3.13/1.74 thf(func_def_52, type, reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real).
% 3.13/1.74 thf(func_def_53, type, reflec1522834046y_real: poly_poly_real > poly_poly_real).
% 3.13/1.74 thf(func_def_54, type, reflect_poly_real: poly_real > poly_real).
% 3.13/1.74 thf(func_def_55, type, collect_real: (real > $o) > set_real).
% 3.13/1.74 thf(func_def_56, type, sturm_1076696862f_real: poly_real > real).
% 3.13/1.74 thf(func_def_57, type, sturm_1308388506f_real: poly_real > real).
% 3.13/1.74 thf(func_def_58, type, member_real: real > set_real > $o).
% 3.13/1.74 thf(func_def_59, type, lb1: real).
% 3.13/1.74 thf(func_def_60, type, lb2: real).
% 3.13/1.74 thf(func_def_61, type, lb: real).
% 3.13/1.74 thf(func_def_62, type, p: poly_real).
% 3.13/1.74 thf(func_def_78, type, sP0: real > real > real > $o).
% 3.13/1.74 thf(func_def_79, type, sP1: real > real > real > $o).
% 3.13/1.74 thf(func_def_80, type, sP2: real > real > real > $o).
% 3.13/1.74 thf(func_def_81, type, sP3: nat > nat > nat > $o).
% 3.13/1.74 thf(func_def_82, type, sP4: nat > nat > nat > $o).
% 3.13/1.74 thf(func_def_83, type, sP5: nat > nat > nat > $o).
% 3.13/1.74 thf(func_def_84, type, sK6: (real > nat) > real).
% 3.13/1.74 thf(func_def_85, type, sK7: (real > nat) > real).
% 3.13/1.74 thf(func_def_86, type, sK8: poly_real > real > real > real).
% 3.13/1.74 thf(func_def_87, type, sK9: nat > (nat > $o) > nat).
% 3.13/1.74 thf(func_def_88, type, sK10: (nat > $o) > nat).
% 3.13/1.74 thf(func_def_89, type, sK11: (real > real) > real).
% 3.13/1.74 thf(func_def_90, type, sK12: (real > real) > real).
% 3.13/1.74 thf(func_def_91, type, sK13: (real > real) > real).
% 3.13/1.74 thf(func_def_92, type, sK14: (real > real) > real).
% 3.13/1.74 thf(func_def_93, type, sK15: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_94, type, sK16: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_95, type, sK17: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_96, type, sK18: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_97, type, sK19: real > poly_real > real).
% 3.13/1.74 thf(func_def_98, type, sK20: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_99, type, sK21: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_100, type, sK22: (real > real) > real).
% 3.13/1.74 thf(func_def_101, type, sK23: (real > real) > real).
% 3.13/1.74 thf(func_def_102, type, sK24: real > real > real).
% 3.13/1.74 thf(func_def_103, type, sK25: (real > real) > real).
% 3.13/1.74 thf(func_def_104, type, sK26: (real > real) > real).
% 3.13/1.74 thf(func_def_105, type, sK27: (nat > real) > nat).
% 3.13/1.74 thf(func_def_106, type, sK28: (nat > real) > nat).
% 3.13/1.74 thf(func_def_107, type, sK29: (nat > nat > $o) > nat).
% 3.13/1.74 thf(func_def_108, type, sK30: (nat > nat > $o) > nat).
% 3.13/1.74 thf(func_def_109, type, sK31: (nat > nat > $o) > nat).
% 3.13/1.74 thf(func_def_110, type, sK32: (nat > nat > $o) > nat).
% 3.13/1.74 thf(func_def_111, type, sK33: (nat > nat > $o) > nat).
% 3.13/1.74 thf(func_def_112, type, sK34: (nat > real) > nat).
% 3.13/1.74 thf(func_def_113, type, sK35: (nat > real) > nat).
% 3.13/1.74 thf(func_def_114, type, sK36: (nat > $o) > nat).
% 3.13/1.74 thf(func_def_115, type, sK37: poly_real > real).
% 3.13/1.74 thf(func_def_116, type, sK38: (real > real) > real).
% 3.13/1.74 thf(func_def_117, type, sK39: (real > real) > real).
% 3.13/1.74 thf(func_def_118, type, sK40: (nat > real) > nat).
% 3.13/1.74 thf(func_def_119, type, sK41: (nat > real) > nat).
% 3.13/1.74 thf(func_def_120, type, sK42: (real > nat) > real).
% 3.13/1.74 thf(func_def_121, type, sK43: (real > nat) > real).
% 3.13/1.74 thf(func_def_122, type, sK44: real).
% 3.13/1.74 thf(func_def_123, type, sK45: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_124, type, sK46: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_125, type, sK47: (nat > real) > nat).
% 3.13/1.74 thf(func_def_126, type, sK48: (nat > real) > nat).
% 3.13/1.74 thf(func_def_127, type, sK49: (real > real) > real).
% 3.13/1.74 thf(func_def_128, type, sK50: (real > real) > real).
% 3.13/1.74 thf(func_def_129, type, sK51: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_130, type, sK52: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_131, type, sK53: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_132, type, sK54: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_133, type, sK55: (real > nat) > real).
% 3.13/1.74 thf(func_def_134, type, sK56: (real > nat) > real).
% 3.13/1.74 thf(func_def_135, type, sK57: (real > real) > real).
% 3.13/1.74 thf(func_def_136, type, sK58: (real > real) > real).
% 3.13/1.74 thf(func_def_137, type, sK59: (real > nat) > real).
% 3.13/1.74 thf(func_def_138, type, sK60: (real > nat) > real).
% 3.13/1.74 thf(func_def_139, type, sK61: (nat > real) > nat).
% 3.13/1.74 thf(func_def_140, type, sK62: (nat > real) > nat).
% 3.13/1.74 thf(func_def_141, type, sK63: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_142, type, sK64: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_143, type, sK65: (real > nat) > real).
% 3.13/1.74 thf(func_def_144, type, sK66: (real > nat) > real).
% 3.13/1.74 thf(func_def_145, type, sK67: (real > real > $o) > real).
% 3.13/1.74 thf(func_def_146, type, sK68: (real > real > $o) > real).
% 3.13/1.74 thf(func_def_147, type, sK69: (real > real > $o) > real).
% 3.13/1.74 thf(func_def_148, type, sK70: (real > real > $o) > real).
% 3.13/1.74 thf(func_def_149, type, sK71: (real > real > $o) > real).
% 3.13/1.74 thf(func_def_150, type, sK72: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_151, type, sK73: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_152, type, sK74: (real > nat) > real).
% 3.13/1.74 thf(func_def_153, type, sK75: (real > nat) > real).
% 3.13/1.74 thf(func_def_154, type, sK76: (real > real) > real).
% 3.13/1.74 thf(func_def_155, type, sK77: (real > real) > real).
% 3.13/1.74 thf(func_def_156, type, sK78: (nat > real) > nat).
% 3.13/1.74 thf(func_def_157, type, sK79: (nat > real) > nat).
% 3.13/1.74 thf(func_def_158, type, sK80: (real > real) > real).
% 3.13/1.74 thf(func_def_159, type, sK81: (real > real) > real).
% 3.13/1.74 thf(func_def_160, type, sK82: (real > nat) > real).
% 3.13/1.74 thf(func_def_161, type, sK83: (real > nat) > real).
% 3.13/1.74 thf(func_def_162, type, sK84: (real > real > $o) > real).
% 3.13/1.74 thf(func_def_163, type, sK85: (real > real > $o) > real).
% 3.13/1.74 thf(func_def_164, type, sK86: (real > real > $o) > real).
% 3.13/1.74 thf(func_def_165, type, sK87: (real > real > $o) > real).
% 3.13/1.74 thf(func_def_166, type, sK88: (nat > real) > nat).
% 3.13/1.74 thf(func_def_167, type, sK89: (nat > real) > nat).
% 3.13/1.74 thf(func_def_168, type, sK90: (nat > nat > $o) > nat).
% 3.13/1.74 thf(func_def_169, type, sK91: (nat > nat > $o) > nat).
% 3.13/1.74 thf(func_def_170, type, sK92: (nat > nat > $o) > nat).
% 3.13/1.74 thf(func_def_171, type, sK93: (nat > nat > $o) > nat).
% 3.13/1.74 thf(func_def_172, type, sK94: real > real > poly_real > real).
% 3.13/1.74 thf(func_def_173, type, sK95: real > real > poly_real > real).
% 3.13/1.74 thf(func_def_174, type, sK96: real > real > poly_real > real).
% 3.13/1.74 thf(func_def_175, type, sK97: real > real).
% 3.13/1.74 thf(func_def_176, type, sK98: nat > (nat > $o) > nat).
% 3.13/1.74 thf(func_def_177, type, sK99: nat > (nat > $o) > nat).
% 3.13/1.74 thf(func_def_178, type, sK100: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_179, type, sK101: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_180, type, sK102: (real > real) > real).
% 3.13/1.74 thf(func_def_181, type, sK103: (real > real) > real).
% 3.13/1.74 thf(func_def_182, type, sK104: poly_real > real > real).
% 3.13/1.74 thf(func_def_183, type, sK105: (nat > real) > nat).
% 3.13/1.74 thf(func_def_184, type, sK106: (nat > real) > nat).
% 3.13/1.74 thf(func_def_185, type, sK107: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_186, type, sK108: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_187, type, sK109: nat > nat).
% 3.13/1.74 thf(func_def_188, type, sK110: (nat > real) > nat).
% 3.13/1.74 thf(func_def_189, type, sK111: (nat > real) > nat).
% 3.13/1.74 thf(func_def_190, type, sK112: (nat > real) > nat).
% 3.13/1.74 thf(func_def_191, type, sK113: (nat > real) > nat).
% 3.13/1.74 thf(func_def_192, type, sK114: (nat > $o) > nat).
% 3.13/1.74 thf(func_def_193, type, sK115: real > real > poly_real > real).
% 3.13/1.74 thf(func_def_194, type, sK116: real > real > real).
% 3.13/1.74 thf(func_def_195, type, sK117: (nat > $o) > nat).
% 3.13/1.74 thf(func_def_196, type, sK118: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_197, type, sK119: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_198, type, sK120: (real > nat) > real).
% 3.13/1.74 thf(func_def_199, type, sK121: (real > nat) > real).
% 3.13/1.74 thf(func_def_200, type, sK122: (real > nat) > real).
% 3.13/1.74 thf(func_def_201, type, sK123: (real > nat) > real).
% 3.13/1.74 thf(func_def_202, type, sK124: real > real).
% 3.13/1.74 thf(func_def_203, type, sK125: (nat > real) > nat).
% 3.13/1.74 thf(func_def_204, type, sK126: (nat > real) > nat).
% 3.13/1.74 thf(func_def_205, type, sK127: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_206, type, sK128: (nat > nat) > nat).
% 3.13/1.74 thf(func_def_207, type, sK129: (real > nat) > real).
% 3.13/1.74 thf(func_def_208, type, sK130: (real > nat) > real).
% 3.13/1.74 thf(func_def_209, type, sK131: real > poly_real > real > real).
% 3.13/1.74 thf(func_def_210, type, sK132: real > poly_real > real > real).
% 3.13/1.74 thf(func_def_211, type, sK133: real > poly_real > real > real).
% 3.13/1.74 thf(func_def_212, type, sK134: (real > nat) > real).
% 3.13/1.74 thf(func_def_213, type, sK135: (real > nat) > real).
% 3.13/1.74 thf(func_def_214, type, sK136: real).
% 3.13/1.74 thf(func_def_216, type, ph138: !>[X0: $tType]:(X0)).
% 3.13/1.74 thf(f2107,plain,(
% 3.13/1.74 $false),
% 3.13/1.74 inference(subsumption_resolution,[status(thm)],[f2105,f2009])).
% 3.13/1.74 thf(f2009,plain,(
% 3.13/1.74 ((ord_less_real @ lb1 @ sK136) != $true)),
% 3.13/1.74 inference(trivial_inequality_removal,[status(thm)],[f1987])).
% 3.13/1.74 thf(f1987,plain,(
% 3.13/1.74 ((ord_less_real @ lb1 @ sK136) != $true) | ($true != $true)),
% 3.13/1.74 inference(superposition,[status(thm)],[f1853,f1767])).
% 3.13/1.74 thf(f1767,plain,(
% 3.13/1.74 ( ! [X2 : real,X0 : real,X1 : real] : (((ord_less_real @ (ord_min_real @ X1 @ X2) @ X0) = $true) | ((ord_less_real @ X1 @ X0) != $true)) )),
% 3.13/1.74 inference(cnf_transformation,[status(thm)],[f1191])).
% 3.13/1.74 thf(f1191,plain,(
% 3.13/1.74 ! [X0 : real,X1 : real,X2 : real] : (((ord_less_real @ X1 @ X0) != $true) | ((ord_less_real @ (ord_min_real @ X1 @ X2) @ X0) = $true))),
% 3.13/1.74 inference(ennf_transformation,[status(thm)],[f955])).
% 3.13/1.74 thf(f955,plain,(
% 3.13/1.74 ! [X0 : real,X1 : real,X2 : real] : (((ord_less_real @ X1 @ X0) = $true) => ((ord_less_real @ (ord_min_real @ X1 @ X2) @ X0) = $true))),
% 3.13/1.74 inference(fool_elimination,[status(thm)],[f954])).
% 3.13/1.74 thf(f954,plain,(
% 3.13/1.74 ! [X0 : real,X1 : real,X2 : real] : ((ord_less_real @ X1 @ X0) => (ord_less_real @ (ord_min_real @ X1 @ X2) @ X0))),
% 3.13/1.74 inference(rectify,[status(thm)],[f42])).
% 3.13/1.74 thf(f42,axiom,(
% 3.13/1.74 ! [X16 : real,X4 : real,X5 : real] : ((ord_less_real @ X4 @ X16) => (ord_less_real @ (ord_min_real @ X4 @ X5) @ X16))),
% 3.13/1.74 file('/export/starexec/sandbox/tmp/tmp.iTAcLxWr78/DTF2THF_18417.p',fact_41_min_Ostrict__coboundedI1)).
% 3.13/1.74 thf(f1853,plain,(
% 3.13/1.74 ((ord_less_real @ (ord_min_real @ lb1 @ lb2) @ sK136) != $true)),
% 3.13/1.74 inference(definition_unfolding,[status(thm)],[f1813,f1530])).
% 3.13/1.74 thf(f1530,plain,(
% 3.13/1.74 (lb = (ord_min_real @ lb1 @ lb2))),
% 3.13/1.74 inference(cnf_transformation,[status(thm)],[f2])).
% 3.13/1.74 thf(f2,axiom,(
% 3.13/1.74 (lb = (ord_min_real @ lb1 @ lb2))),
% 3.13/1.74 file('/export/starexec/sandbox/tmp/tmp.iTAcLxWr78/DTF2THF_18417.p',fact_1_lb__def)).
% 3.13/1.74 thf(f1813,plain,(
% 3.13/1.74 ((ord_less_real @ lb @ sK136) != $true)),
% 3.13/1.74 inference(cnf_transformation,[status(thm)],[f1464])).
% 3.13/1.74 thf(f1464,plain,(
% 3.13/1.74 (zero_zero_real = (poly_real2 @ p @ sK136)) & ((ord_less_real @ lb @ sK136) != $true)),
% 3.13/1.74 inference(skolemisation,[status(esa),new_symbols(skolem,[sK136])],[f1209,f1463])).
% 3.13/1.74 thf(f1463,plain,(
% 3.13/1.74 ? [X0 : real] : (((poly_real2 @ p @ X0) = zero_zero_real) & ((ord_less_real @ lb @ X0) != $true)) => ((zero_zero_real = (poly_real2 @ p @ sK136)) & ((ord_less_real @ lb @ sK136) != $true))),
% 3.13/1.74 introduced(definition,[],[choice_axiom])).
% 3.13/1.74 thf(f1209,plain,(
% 3.13/1.74 ? [X0 : real] : (((poly_real2 @ p @ X0) = zero_zero_real) & ((ord_less_real @ lb @ X0) != $true))),
% 3.13/1.74 inference(ennf_transformation,[status(thm)],[f451])).
% 3.13/1.74 thf(f451,plain,(
% 3.13/1.74 ~! [X0 : real] : (((poly_real2 @ p @ X0) != zero_zero_real) | ((ord_less_real @ lb @ X0) = $true))),
% 3.13/1.74 inference(fool_elimination,[status(thm)],[f450])).
% 3.13/1.74 thf(f450,plain,(
% 3.13/1.74 ~! [X0 : real] : (((poly_real2 @ p @ X0) != zero_zero_real) | (ord_less_real @ lb @ X0))),
% 3.13/1.74 inference(rectify,[status(thm)],[f352])).
% 3.13/1.74 thf(f352,negated_conjecture,(
% 3.13/1.74 ~! [X7 : real] : ((zero_zero_real != (poly_real2 @ p @ X7)) | (ord_less_real @ lb @ X7))),
% 3.13/1.74 inference(negated_conjecture,[status(cth)],[f351])).
% 3.13/1.74 thf(f351,conjecture,(
% 3.13/1.74 ! [X7 : real] : ((zero_zero_real != (poly_real2 @ p @ X7)) | (ord_less_real @ lb @ X7))),
% 3.13/1.74 file('/export/starexec/sandbox/tmp/tmp.iTAcLxWr78/DTF2THF_18417.p',conj_0)).
% 3.13/1.74 thf(f2105,plain,(
% 3.13/1.74 ((ord_less_real @ lb1 @ sK136) = $true)),
% 3.13/1.74 inference(equality_resolution,[status(thm)],[f1837])).
% 3.13/1.74 thf(f1837,plain,(
% 3.13/1.74 ( ! [X0 : real] : (((poly_real2 @ p @ X0) != (poly_real2 @ p @ sK136)) | ((ord_less_real @ lb1 @ X0) = $true)) )),
% 3.13/1.74 inference(definition_unfolding,[status(thm)],[f1752,f1814])).
% 3.13/1.74 thf(f1814,plain,(
% 3.13/1.74 (zero_zero_real = (poly_real2 @ p @ sK136))),
% 3.13/1.74 inference(cnf_transformation,[status(thm)],[f1464])).
% 3.13/1.74 thf(f1752,plain,(
% 3.13/1.74 ( ! [X0 : real] : (((ord_less_real @ lb1 @ X0) = $true) | ((poly_real2 @ p @ X0) != zero_zero_real)) )),
% 3.13/1.74 inference(cnf_transformation,[status(thm)],[f1054])).
% 3.13/1.74 thf(f1054,plain,(
% 3.13/1.74 ! [X0 : real] : (((ord_less_real @ lb1 @ X0) = $true) | ((poly_real2 @ p @ X0) != zero_zero_real))),
% 3.13/1.74 inference(ennf_transformation,[status(thm)],[f447])).
% 3.13/1.74 thf(f447,plain,(
% 3.13/1.74 ! [X0 : real] : (((poly_real2 @ p @ X0) = zero_zero_real) => ((ord_less_real @ lb1 @ X0) = $true))),
% 3.13/1.74 inference(fool_elimination,[status(thm)],[f446])).
% 3.13/1.74 thf(f446,plain,(
% 3.13/1.74 ! [X0 : real] : (((poly_real2 @ p @ X0) = zero_zero_real) => (ord_less_real @ lb1 @ X0))),
% 3.13/1.74 inference(rectify,[status(thm)],[f1])).
% 3.13/1.74 thf(f1,axiom,(
% 3.13/1.74 ! [X0 : real] : (((poly_real2 @ p @ X0) = zero_zero_real) => (ord_less_real @ lb1 @ X0))),
% 3.13/1.74 file('/export/starexec/sandbox/tmp/tmp.iTAcLxWr78/DTF2THF_18417.p',fact_0_lb1)).
% 3.13/1.74 % SZS output end Proof for DTF2THF_18417
% 3.13/1.74 % (18536)------------------------------
% 3.13/1.74 % (18536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.74 % (18536)Termination reason: Refutation
% 3.13/1.74
% 3.13/1.74 % (18536)Memory used [KB]: 7164
% 3.13/1.74 % (18536)Time elapsed: 0.070 s
% 3.13/1.74 % (18536)Instructions burned: 204 (million)
% 3.13/1.74 % (18536)------------------------------
% 3.13/1.74 % (18536)------------------------------
% 3.13/1.74 % (18530)Success in time 0.087 s
%------------------------------------------------------------------------------