%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP094^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:30 AM UTC 2026
% Result : Theorem 2.58s 1.93s
% Output : Refutation 2.58s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 7
% Syntax : Number of formulae : 52 ( 21 unt; 0 typ; 2 def)
% Number of atoms : 166 ( 108 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 381 ( 38 ~; 35 |; 0 &; 296 @)
% ( 2 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 168 ( 164 usr; 27 con; 0-6 aty)
% ( 3 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 44 ( 27 ^; 17 !; 0 ?; 44 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
real: $tType ).
thf(type_def_7,type,
set_nat: $tType ).
thf(type_def_8,type,
set_poly_poly_real: $tType ).
thf(type_def_9,type,
set_poly_real: $tType ).
thf(type_def_10,type,
set_real: $tType ).
thf(type_def_11,type,
nat: $tType ).
thf(type_def_12,type,
poly_nat: $tType ).
thf(type_def_13,type,
poly_poly_real: $tType ).
thf(type_def_14,type,
poly_real: $tType ).
thf(type_def_15,type,
poly_poly_nat: $tType ).
thf(type_def_16,type,
poly_poly_poly_real: $tType ).
thf(func_def_0,type,
poly_poly_poly_real: $tType ).
thf(func_def_1,type,
set_poly_poly_real: $tType ).
thf(func_def_2,type,
poly_poly_real: $tType ).
thf(func_def_3,type,
poly_poly_nat: $tType ).
thf(func_def_4,type,
set_poly_real: $tType ).
thf(func_def_5,type,
poly_real: $tType ).
thf(func_def_6,type,
poly_nat: $tType ).
thf(func_def_7,type,
set_real: $tType ).
thf(func_def_8,type,
set_nat: $tType ).
thf(func_def_9,type,
real: $tType ).
thf(func_def_10,type,
nat: $tType ).
thf(func_def_11,type,
inverse_inverse_real: real > real ).
thf(func_def_12,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_13,type,
finite1328464339y_real: set_poly_poly_real > $o ).
thf(func_def_14,type,
finite1810960971y_real: set_poly_real > $o ).
thf(func_def_15,type,
finite_finite_real: set_real > $o ).
thf(func_def_16,type,
one_one_nat: nat ).
thf(func_def_17,type,
one_one_poly_nat: poly_nat ).
thf(func_def_18,type,
one_on501200385y_real: poly_poly_real ).
thf(func_def_19,type,
one_one_poly_real: poly_real ).
thf(func_def_20,type,
one_one_real: real ).
thf(func_def_21,type,
zero_zero_nat: nat ).
thf(func_def_22,type,
zero_zero_poly_nat: poly_nat ).
thf(func_def_23,type,
zero_z1059985641ly_nat: poly_poly_nat ).
thf(func_def_24,type,
zero_z935034829y_real: poly_poly_poly_real ).
thf(func_def_25,type,
zero_z1423781445y_real: poly_poly_real ).
thf(func_def_26,type,
zero_zero_poly_real: poly_real ).
thf(func_def_27,type,
zero_zero_real: real ).
thf(func_def_28,type,
ring_1897377867y_real: set_poly_poly_real ).
thf(func_def_29,type,
ring_1690226883y_real: set_poly_real ).
thf(func_def_30,type,
ring_1_Ints_real: set_real ).
thf(func_def_31,type,
suc: nat > nat ).
thf(func_def_32,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_33,type,
ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
thf(func_def_34,type,
ord_less_poly_real: poly_real > poly_real > $o ).
thf(func_def_35,type,
ord_less_real: real > real > $o ).
thf(func_def_36,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_37,type,
algebraic_real: real > $o ).
thf(func_def_38,type,
algebraic_int_real: real > $o ).
thf(func_def_39,type,
cr_poly_real: ( nat > real ) > poly_real > $o ).
thf(func_def_40,type,
degree_nat: poly_nat > nat ).
thf(func_def_41,type,
degree_poly_nat: poly_poly_nat > nat ).
thf(func_def_42,type,
degree360860553y_real: poly_poly_poly_real > nat ).
thf(func_def_43,type,
degree_poly_real: poly_poly_real > nat ).
thf(func_def_44,type,
degree_real: poly_real > nat ).
thf(func_def_45,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_46,type,
divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real ).
thf(func_def_47,type,
divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real ).
thf(func_def_48,type,
is_zero_nat: poly_nat > $o ).
thf(func_def_49,type,
is_zero_poly_real: poly_poly_real > $o ).
thf(func_def_50,type,
is_zero_real: poly_real > $o ).
thf(func_def_51,type,
order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
thf(func_def_52,type,
order_poly_real: poly_real > poly_poly_real > nat ).
thf(func_def_53,type,
order_real: real > poly_real > nat ).
thf(func_def_54,type,
pcr_poly_nat_nat: ( nat > nat > $o ) > ( nat > nat ) > poly_nat > $o ).
thf(func_def_55,type,
pcr_po273983709ly_nat: ( poly_nat > poly_nat > $o ) > ( nat > poly_nat ) > poly_poly_nat > $o ).
thf(func_def_56,type,
pcr_po1200519205y_real: ( poly_poly_real > poly_poly_real > $o ) > ( nat > poly_poly_real ) > poly_poly_poly_real > $o ).
thf(func_def_57,type,
pcr_po1314690837y_real: ( poly_real > poly_real > $o ) > ( nat > poly_real ) > poly_poly_real > $o ).
thf(func_def_58,type,
pcr_poly_real_real: ( real > real > $o ) > ( nat > real ) > poly_real > $o ).
thf(func_def_59,type,
pderiv_nat: poly_nat > poly_nat ).
thf(func_def_60,type,
pderiv_poly_real: poly_poly_real > poly_poly_real ).
thf(func_def_61,type,
pderiv_real: poly_real > poly_real ).
thf(func_def_62,type,
poly_nat2: poly_nat > nat > nat ).
thf(func_def_63,type,
poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
thf(func_def_64,type,
poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
thf(func_def_65,type,
poly_poly_real2: poly_poly_real > poly_real > poly_real ).
thf(func_def_66,type,
poly_real2: poly_real > real > real ).
thf(func_def_67,type,
coeff_nat: poly_nat > nat > nat ).
thf(func_def_68,type,
coeff_poly_nat: poly_poly_nat > nat > poly_nat ).
thf(func_def_69,type,
coeff_poly_poly_real: poly_poly_poly_real > nat > poly_poly_real ).
thf(func_def_70,type,
coeff_poly_real: poly_poly_real > nat > poly_real ).
thf(func_def_71,type,
coeff_real: poly_real > nat > real ).
thf(func_def_72,type,
poly_cutoff_nat: nat > poly_nat > poly_nat ).
thf(func_def_73,type,
poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
thf(func_def_74,type,
poly_cutoff_real: nat > poly_real > poly_real ).
thf(func_def_75,type,
poly_shift_nat: nat > poly_nat > poly_nat ).
thf(func_def_76,type,
poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
thf(func_def_77,type,
poly_shift_real: nat > poly_real > poly_real ).
thf(func_def_78,type,
reflect_poly_nat: poly_nat > poly_nat ).
thf(func_def_79,type,
reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
thf(func_def_80,type,
reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
thf(func_def_81,type,
reflec1522834046y_real: poly_poly_real > poly_poly_real ).
thf(func_def_82,type,
reflect_poly_real: poly_real > poly_real ).
thf(func_def_83,type,
rsquar1555552848y_real: poly_poly_real > $o ).
thf(func_def_84,type,
rsquarefree_real: poly_real > $o ).
thf(func_def_85,type,
synthetic_div_nat: poly_nat > nat > poly_nat ).
thf(func_def_86,type,
synthe1498897281y_real: poly_poly_real > poly_real > poly_poly_real ).
thf(func_def_87,type,
synthetic_div_real: poly_real > real > poly_real ).
thf(func_def_88,type,
field_1537545994s_real: set_real ).
thf(func_def_89,type,
dvd_dvd_nat: nat > nat > $o ).
thf(func_def_90,type,
dvd_dvd_poly_nat: poly_nat > poly_nat > $o ).
thf(func_def_91,type,
dvd_dv1946063458y_real: poly_poly_real > poly_poly_real > $o ).
thf(func_def_92,type,
dvd_dvd_poly_real: poly_real > poly_real > $o ).
thf(func_def_93,type,
dvd_dvd_real: real > real > $o ).
thf(func_def_94,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_95,type,
collec927113489y_real: ( poly_poly_real > $o ) > set_poly_poly_real ).
thf(func_def_96,type,
collect_poly_real: ( poly_real > $o ) > set_poly_real ).
thf(func_def_97,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_98,type,
member_nat: nat > set_nat > $o ).
thf(func_def_99,type,
member1159720147y_real: poly_poly_real > set_poly_poly_real > $o ).
thf(func_def_100,type,
member_poly_real: poly_real > set_poly_real > $o ).
thf(func_def_101,type,
member_real: real > set_real > $o ).
thf(func_def_102,type,
n: nat ).
thf(func_def_103,type,
p: poly_real ).
thf(func_def_104,type,
x: real ).
thf(func_def_122,type,
sK1: ( nat > real ) > nat ).
thf(func_def_123,type,
sK2: real > real ).
thf(func_def_124,type,
sK3: set_real > ( nat > real > $o ) > set_nat > real ).
thf(func_def_125,type,
sK4: set_real > ( nat > real > $o ) > set_nat > nat ).
thf(func_def_126,type,
sK5: poly_real ).
thf(func_def_127,type,
sK6: nat > nat ).
thf(func_def_128,type,
sK7: ( real > poly_real > $o ) > set_real > set_poly_real > real ).
thf(func_def_129,type,
sK8: ( real > poly_real > $o ) > set_real > set_poly_real > poly_real ).
thf(func_def_130,type,
sK9: ( nat > $o ) > nat ).
thf(func_def_131,type,
sK10: ( poly_real > $o ) > poly_real ).
thf(func_def_132,type,
sK11: set_real > ( poly_real > real > $o ) > set_poly_real > real ).
thf(func_def_133,type,
sK12: set_real > ( poly_real > real > $o ) > set_poly_real > poly_real ).
thf(func_def_134,type,
sK13: nat > nat > nat ).
thf(func_def_135,type,
sK14: nat > nat > nat ).
thf(func_def_136,type,
sK15: real > real > real ).
thf(func_def_137,type,
sK16: poly_real > nat ).
thf(func_def_138,type,
sK17: poly_poly_real > nat > nat ).
thf(func_def_139,type,
sK18: set_poly_real > ( poly_real > poly_real > $o ) > set_poly_real > poly_real ).
thf(func_def_140,type,
sK19: real > real ).
thf(func_def_141,type,
sK20: ( nat > $o ) > nat > nat ).
thf(func_def_142,type,
sK21: ( nat > $o ) > nat > nat ).
thf(func_def_143,type,
sK22: poly_real ).
thf(func_def_144,type,
sK23: ( nat > $o ) > nat ).
thf(func_def_145,type,
sK24: poly_real > poly_real > nat ).
thf(func_def_146,type,
sK25: set_nat > ( nat > poly_real > $o ) > set_poly_real > nat ).
thf(func_def_147,type,
sK26: ( nat > $o ) > nat ).
thf(func_def_148,type,
sK27: ( real > nat > $o ) > set_nat > set_real > real ).
thf(func_def_149,type,
sK28: ( real > nat > $o ) > set_nat > set_real > nat ).
thf(func_def_150,type,
sK29: poly_real > real ).
thf(func_def_151,type,
sK30: real > poly_real > real > real ).
thf(func_def_152,type,
sK31: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_153,type,
sK32: nat > ( nat > $o ) > nat ).
thf(func_def_154,type,
sK33: ( nat > $o ) > nat ).
thf(func_def_155,type,
sK34: ( poly_real > $o ) > ( poly_real > $o ) > poly_real ).
thf(func_def_156,type,
sK35: nat > ( nat > $o ) > nat ).
thf(func_def_157,type,
sK36: nat > ( nat > $o ) > nat ).
thf(func_def_158,type,
sK37: ( nat > nat > $o ) > nat ).
thf(func_def_159,type,
sK38: ( nat > nat > $o ) > nat ).
thf(func_def_160,type,
sK39: ( nat > nat > $o ) > nat ).
thf(func_def_161,type,
sK40: ( nat > nat > $o ) > nat ).
thf(func_def_162,type,
sK41: real > poly_real ).
thf(func_def_163,type,
sK42: ( nat > $o ) > nat ).
thf(func_def_164,type,
sK43: poly_poly_real > poly_real ).
thf(func_def_165,type,
sK44: ( nat > $o ) > nat > nat ).
thf(func_def_166,type,
sK45: ( nat > $o ) > nat > nat ).
thf(func_def_167,type,
sK46: ( nat > nat > $o ) > nat ).
thf(func_def_168,type,
sK47: ( nat > nat > $o ) > nat ).
thf(func_def_169,type,
sK48: ( nat > nat > $o ) > nat ).
thf(func_def_170,type,
sK49: ( nat > nat > $o ) > nat ).
thf(func_def_171,type,
sK50: ( real > $o ) > real ).
thf(func_def_172,type,
sK51: ( real > $o ) > poly_real ).
thf(func_def_173,type,
sK52: ( real > $o ) > ( real > $o ) > real ).
thf(func_def_174,type,
sK53: ( nat > $o ) > nat ).
thf(func_def_175,type,
sK54: nat > ( nat > $o ) > nat ).
thf(func_def_176,type,
sK55: real > poly_real ).
thf(func_def_177,type,
sK56: poly_real > nat ).
thf(func_def_178,type,
sK57: nat > nat > nat ).
thf(func_def_179,type,
sK58: ( nat > $o ) > nat ).
thf(func_def_180,type,
sK59: ( real > $o ) > real ).
thf(func_def_181,type,
sK60: set_nat > ( poly_real > nat > $o ) > set_poly_real > poly_real ).
thf(func_def_182,type,
sK61: nat > nat ).
thf(func_def_183,type,
sK62: ( nat > nat ) > nat ).
thf(func_def_184,type,
sK63: set_nat > ( nat > nat > $o ) > set_nat > nat ).
thf(func_def_185,type,
sK64: set_real > set_real > ( real > real > $o ) > real ).
thf(func_def_186,type,
ph65:
!>[X0: $tType] : X0 ).
thf(func_def_187,type,
sK66: set_real > real ).
thf(func_def_188,type,
sK67: real ).
thf(func_def_189,type,
sK68: poly_real > real ).
thf(f3552,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3517,f3531,f3551]) ).
thf(f3551,plain,
~ spl0_9,
inference(avatar_contradiction_clause,[status(thm)],[f3550]) ).
thf(f3550,plain,
( $false
| ~ spl0_9 ),
inference(trivial_inequality_removal,[status(thm)],[f3549]) ).
thf(f3549,plain,
( ( ( pderiv_real @ one_one_poly_real )
!= ( pderiv_real @ one_one_poly_real ) )
| ~ spl0_9 ),
inference(superposition,[status(thm)],[f1272,f3514]) ).
thf(f3514,plain,
( ( ( pderiv_real @ p )
= ( pderiv_real @ one_one_poly_real ) )
| ~ spl0_9 ),
inference(avatar_component_clause,[status(thm)],[f3512]) ).
thf(f3512,definition,
( spl0_9
<=> ( ( pderiv_real @ p )
= ( pderiv_real @ one_one_poly_real ) ) ),
introduced(definition,[new_symbols(naming,[spl0_9])],[avatar_definition]) ).
thf(f1272,plain,
( ( pderiv_real @ p )
!= ( pderiv_real @ one_one_poly_real ) ),
inference(definition_unfolding,[status(thm)],[f1145,f1081]) ).
thf(f1081,plain,
( zero_zero_poly_real
= ( pderiv_real @ one_one_poly_real ) ),
inference(cnf_transformation,[status(thm)],[f855]) ).
thf(f855,plain,
( zero_zero_poly_real
= ( pderiv_real @ one_one_poly_real ) ),
inference(fool_elimination,[status(thm)],[f172]) ).
thf(f172,axiom,
( zero_zero_poly_real
= ( pderiv_real @ one_one_poly_real ) ),
file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',fact_171_pderiv__1) ).
thf(f1145,plain,
( zero_zero_poly_real
!= ( pderiv_real @ p ) ),
inference(cnf_transformation,[status(thm)],[f896]) ).
thf(f896,plain,
( zero_zero_poly_real
!= ( pderiv_real @ p ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,axiom,
( zero_zero_poly_real
!= ( pderiv_real @ p ) ),
file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',fact_4__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062) ).
thf(f3531,plain,
( spl0_9
| ~ spl0_8 ),
inference(avatar_split_clause,[status(thm)],[f3530,f3508,f3512]) ).
thf(f3508,definition,
( spl0_8
<=> ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ ( pderiv_real @ p ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl0_8])],[avatar_definition]) ).
thf(f3530,plain,
( ( ( pderiv_real @ p )
= ( pderiv_real @ one_one_poly_real ) )
| ~ spl0_8 ),
inference(trivial_inequality_removal,[status(thm)],[f3529]) ).
thf(f3529,plain,
( ( ( poly_real2 @ p @ x )
!= ( poly_real2 @ p @ x ) )
| ( ( pderiv_real @ p )
= ( pderiv_real @ one_one_poly_real ) )
| ~ spl0_8 ),
inference(duplicate_literal_removal,[status(thm)],[f3528]) ).
thf(f3528,plain,
( ( ( pderiv_real @ p )
= ( pderiv_real @ one_one_poly_real ) )
| ( ( poly_real2 @ p @ x )
!= ( poly_real2 @ p @ x ) )
| ( ( poly_real2 @ p @ x )
!= ( poly_real2 @ p @ x ) )
| ~ spl0_8 ),
inference(superposition,[status(thm)],[f3490,f3510]) ).
thf(f3510,plain,
( ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ ( pderiv_real @ p ) ) ) )
| ~ spl0_8 ),
inference(avatar_component_clause,[status(thm)],[f3508]) ).
thf(f3490,plain,
! [X0: poly_real] :
( ( ( poly_real2 @ p @ x )
!= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ X0 ) ) )
| ( ( pderiv_real @ one_one_poly_real )
= X0 )
| ( ( poly_real2 @ p @ x )
!= ( poly_real2 @ X0 @ ( sK68 @ X0 ) ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f3489]) ).
thf(f3489,plain,
! [X0: poly_real] :
( ( ( pderiv_real @ one_one_poly_real )
= X0 )
| ( $false
= ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ X0 ) ) ) )
| ( ( poly_real2 @ p @ x )
!= ( poly_real2 @ X0 @ ( sK68 @ X0 ) ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f3488]) ).
thf(f3488,plain,
! [X0: poly_real] :
( ( ( pderiv_real @ one_one_poly_real )
= X0 )
| ( ( ( poly_real2 @ X0 @ ( sK68 @ X0 ) )
= ( poly_real2 @ p @ x ) )
= $false )
| ( $false
= ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ X0 ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f3482]) ).
thf(f3482,plain,
! [X0: poly_real] :
( ( ( pderiv_real @ one_one_poly_real )
= X0 )
| ( ( ( poly_real2 @ X0 @ ( sK68 @ X0 ) )
= ( poly_real2 @ p @ x ) )
!= ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ X0 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f3480]) ).
thf(f3480,plain,
! [X0: poly_real] :
( ( ( pderiv_real @ one_one_poly_real )
= X0 )
| ( ( ^ [Y0: real] :
( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ Y0 ) )
@ ( sK68 @ X0 ) )
!= ( ^ [Y0: real] :
( ( poly_real2 @ X0 @ Y0 )
= ( poly_real2 @ p @ x ) )
@ ( sK68 @ X0 ) ) ) ),
inference(negative_extensionality,[status(thm)],[f3371]) ).
thf(f3371,plain,
! [X0: poly_real] :
( ( ( ^ [Y0: real] :
( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ Y0 ) ) )
!= ( ^ [Y0: real] :
( ( poly_real2 @ X0 @ Y0 )
= ( poly_real2 @ p @ x ) ) ) )
| ( ( pderiv_real @ one_one_poly_real )
= X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f3369]) ).
thf(f3369,plain,
! [X0: poly_real] :
( ( $false = $true )
| ( ( ^ [Y0: real] :
( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ Y0 ) ) )
!= ( ^ [Y0: real] :
( ( poly_real2 @ X0 @ Y0 )
= ( poly_real2 @ p @ x ) ) ) )
| ( ( pderiv_real @ one_one_poly_real )
= X0 ) ),
inference(constrained_superposition,[status(thm)],[f2088,f1413]) ).
thf(f1413,plain,
( ( finite_finite_real
@ ( collect_real
@ ^ [Y0: real] :
( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ Y0 ) ) ) )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f1222]) ).
thf(f1222,plain,
( ( ~ ( finite_finite_real
@ ( collect_real
@ ^ [Y0: real] :
( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ Y0 ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1034,f1078]) ).
thf(f1078,plain,
( zero_zero_real
= ( poly_real2 @ p @ x ) ),
inference(cnf_transformation,[status(thm)],[f409]) ).
thf(f409,plain,
( zero_zero_real
= ( poly_real2 @ p @ x ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,axiom,
( zero_zero_real
= ( poly_real2 @ p @ x ) ),
file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',fact_5_p_I3_J) ).
thf(f1034,plain,
( ( ~ ( finite_finite_real
@ ( collect_real
@ ^ [Y0: real] :
( zero_zero_real
= ( poly_real2 @ ( pderiv_real @ p ) @ Y0 ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f816]) ).
thf(f816,plain,
( ( ~ ( finite_finite_real
@ ( collect_real
@ ^ [Y0: real] :
( zero_zero_real
= ( poly_real2 @ ( pderiv_real @ p ) @ Y0 ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f815]) ).
thf(f815,plain,
~ ( finite_finite_real
@ ( collect_real
@ ^ [X0: real] :
( zero_zero_real
= ( poly_real2 @ ( pderiv_real @ p ) @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f346]) ).
thf(f346,negated_conjecture,
~ ( finite_finite_real
@ ( collect_real
@ ^ [X1: real] :
( zero_zero_real
= ( poly_real2 @ ( pderiv_real @ p ) @ X1 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f345]) ).
thf(f345,conjecture,
( finite_finite_real
@ ( collect_real
@ ^ [X1: real] :
( zero_zero_real
= ( poly_real2 @ ( pderiv_real @ p ) @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',conj_0) ).
thf(f2088,plain,
! [X1: poly_real] :
( ( ( finite_finite_real
@ ( collect_real
@ ^ [Y0: real] :
( ( poly_real2 @ X1 @ Y0 )
= ( poly_real2 @ p @ x ) ) ) )
= $true )
| ( ( pderiv_real @ one_one_poly_real )
= X1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f2087]) ).
thf(f2087,plain,
! [X1: poly_real] :
( ( ( finite_finite_real
@ ( collect_real
@ ^ [Y0: real] :
( ( poly_real2 @ X1 @ Y0 )
= ( poly_real2 @ p @ x ) ) ) )
= $true )
| ( ( ( pderiv_real @ one_one_poly_real )
= X1 )
= $true ) ),
inference(not_proxy_clausification,[status(thm)],[f2086]) ).
thf(f2086,plain,
! [X1: poly_real] :
( ( ( finite_finite_real
@ ( collect_real
@ ^ [Y0: real] :
( ( poly_real2 @ X1 @ Y0 )
= ( poly_real2 @ p @ x ) ) ) )
= $true )
| ( ( ( pderiv_real @ one_one_poly_real )
!= X1 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f2085]) ).
thf(f2085,plain,
! [X1: poly_real] :
( $true
= ( ( ( pderiv_real @ one_one_poly_real )
!= X1 )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Y0: real] :
( ( poly_real2 @ X1 @ Y0 )
= ( poly_real2 @ p @ x ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2084]) ).
thf(f2084,plain,
! [X1: poly_real] :
( ( ^ [Y0: poly_real] :
( ( ( pderiv_real @ one_one_poly_real )
!= Y0 )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Y1: real] :
( ( poly_real2 @ Y0 @ Y1 )
= ( poly_real2 @ p @ x ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1235]) ).
thf(f1235,plain,
( ( !! @ poly_real
@ ^ [Y0: poly_real] :
( ( ( pderiv_real @ one_one_poly_real )
!= Y0 )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Y1: real] :
( ( poly_real2 @ Y0 @ Y1 )
= ( poly_real2 @ p @ x ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1061,f1081,f1078]) ).
thf(f1061,plain,
( ( !! @ poly_real
@ ^ [Y0: poly_real] :
( ( zero_zero_poly_real != Y0 )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Y1: real] :
( ( poly_real2 @ Y0 @ Y1 )
= zero_zero_real ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f614]) ).
thf(f614,plain,
( ( !! @ poly_real
@ ^ [Y0: poly_real] :
( ( zero_zero_poly_real != Y0 )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Y1: real] :
( ( poly_real2 @ Y0 @ Y1 )
= zero_zero_real ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f613]) ).
thf(f613,plain,
! [X0: poly_real] :
( ( zero_zero_poly_real != X0 )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X1: real] :
( ( poly_real2 @ X0 @ X1 )
= zero_zero_real ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X0: poly_real] :
( ( zero_zero_poly_real != X0 )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X1: real] :
( ( poly_real2 @ X0 @ X1 )
= zero_zero_real ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',fact_2_poly__roots__finite) ).
thf(f3517,plain,
( spl0_9
| spl0_8 ),
inference(avatar_split_clause,[status(thm)],[f3502,f3508,f3512]) ).
thf(f3502,plain,
( ( ( pderiv_real @ p )
= ( pderiv_real @ one_one_poly_real ) )
| ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ ( pderiv_real @ p ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3501]) ).
thf(f3501,plain,
( ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ ( pderiv_real @ p ) ) ) )
| ( ( poly_real2 @ p @ x )
!= ( poly_real2 @ p @ x ) )
| ( ( pderiv_real @ p )
= ( pderiv_real @ one_one_poly_real ) ) ),
inference(equality_factoring,[status(thm)],[f3492]) ).
thf(f3492,plain,
! [X0: poly_real] :
( ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ X0 ) ) )
| ( ( poly_real2 @ p @ x )
= ( poly_real2 @ X0 @ ( sK68 @ X0 ) ) )
| ( ( pderiv_real @ one_one_poly_real )
= X0 ) ),
inference(equality_proxy_clausification,[status(thm)],[f3491]) ).
thf(f3491,plain,
! [X0: poly_real] :
( ( ( pderiv_real @ one_one_poly_real )
= X0 )
| ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ X0 ) ) )
| ( ( ( poly_real2 @ X0 @ ( sK68 @ X0 ) )
= ( poly_real2 @ p @ x ) )
= $true ) ),
inference(equality_proxy_clausification,[status(thm)],[f3487]) ).
thf(f3487,plain,
! [X0: poly_real] :
( ( ( pderiv_real @ one_one_poly_real )
= X0 )
| ( ( ( poly_real2 @ p @ x )
= ( poly_real2 @ ( pderiv_real @ p ) @ ( sK68 @ X0 ) ) )
= $true )
| ( ( ( poly_real2 @ X0 @ ( sK68 @ X0 ) )
= ( poly_real2 @ p @ x ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f3482]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP094^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n025.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:26:28 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.47 ---- Original DTF file ---
% 0.29/0.47 thf(spec,logic,$$dhol).
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % File : ITP094^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.47 % Domain : Interactive Theorem Proving
% 0.29/0.47 % Problem : Sledgehammer Liouville_Numbers problem prob_128__5866194_1
% 0.29/0.47 % Version : Especial.
% 0.29/0.47 % English :
% 0.29/0.47
% 0.29/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.47 % Source : [Des21]
% 0.29/0.47 % Names : Liouville_Numbers/prob_128__5866194_1 [Des21]
% 0.29/0.47
% 0.29/0.47 % Status : Theorem
% 0.29/0.47 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% 0.29/0.47 % Syntax : Number of formulae : 450 ( 208 unt; 105 typ; 0 def)
% 0.29/0.47 % Number of atoms : 818 ( 457 equ; 0 cnn)
% 0.29/0.47 % Maximal formula atoms : 7 ( 2 avg)
% 0.29/0.47 % Number of connectives : 1906 ( 173 ~; 17 |; 91 &;1377 @)
% 0.29/0.47 % ( 0 <=>; 248 =>; 0 <=; 0 <~>)
% 0.29/0.47 % Maximal formula depth : 16 ( 5 avg)
% 0.29/0.47 % Number of types : 12 ( 11 usr)
% 0.29/0.47 % Number of type conns : 223 ( 223 >; 0 *; 0 +; 0 <<)
% 0.29/0.47 % Number of symbols : 95 ( 94 usr; 19 con; 0-6 aty)
% 0.29/0.47 % Number of variables : 598 ( 66 ^; 485 !; 47 ?; 598 :)
% 0.29/0.47 % SPC : TH0_THM_EQU_NAR
% 0.29/0.47
% 0.29/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:39:38.922
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % Could-be-implicit typings (11)
% 0.29/0.47 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 0.29/0.47 poly_poly_poly_real: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 0.29/0.47 set_poly_poly_real: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 poly_poly_real: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 0.29/0.47 poly_poly_nat: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 set_poly_real: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 poly_real: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.29/0.47 poly_nat: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/0.47 set_real: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.47 set_nat: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Real__Oreal,type,
% 0.29/0.47 real: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.29/0.47 nat: $tType ).
% 0.29/0.47
% 0.29/0.47 % Explicit typings (94)
% 0.29/0.47 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 0.29/0.47 inverse_inverse_real: real > real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.29/0.47 finite_finite_nat: set_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 finite1328464339y_real: set_poly_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 finite1810960971y_real: set_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
% 0.29/0.47 finite_finite_real: set_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.29/0.47 one_one_nat: nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.29/0.47 one_one_poly_nat: poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 one_on501200385y_real: poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 one_one_poly_real: poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.29/0.47 one_one_real: real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.29/0.47 zero_zero_nat: nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.29/0.47 zero_zero_poly_nat: poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 0.29/0.47 zero_z1059985641ly_nat: poly_poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 0.29/0.47 zero_z935034829y_real: poly_poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 zero_z1423781445y_real: poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 zero_zero_poly_real: poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.29/0.47 zero_zero_real: real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Int_Oring__1__class_OInts_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 ring_1897377867y_real: set_poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Int_Oring__1__class_OInts_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 ring_1690226883y_real: set_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 0.29/0.47 ring_1_Ints_real: set_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Nat_OSuc,type,
% 0.29/0.47 suc: nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.29/0.47 ord_less_nat: nat > nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 ord_less_poly_real: poly_real > poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.29/0.47 ord_less_real: real > real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.29/0.47 ord_less_eq_real: real > real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oalgebraic_001t__Real__Oreal,type,
% 0.29/0.47 algebraic_real: real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oalgebraic__int_001t__Real__Oreal,type,
% 0.29/0.47 algebraic_int_real: real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Ocr__poly_001t__Real__Oreal,type,
% 0.29/0.47 cr_poly_real: ( nat > real ) > poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Odegree_001t__Nat__Onat,type,
% 0.29/0.47 degree_nat: poly_nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.29/0.47 degree_poly_nat: poly_poly_nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 degree360860553y_real: poly_poly_poly_real > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 degree_poly_real: poly_poly_real > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Odegree_001t__Real__Oreal,type,
% 0.29/0.47 degree_real: poly_real > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 divide924636027y_real: poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > nat > nat > poly_poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Odivide__poly__main_001t__Real__Oreal,type,
% 0.29/0.47 divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat,type,
% 0.29/0.47 is_zero_nat: poly_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 is_zero_poly_real: poly_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Ois__zero_001t__Real__Oreal,type,
% 0.29/0.47 is_zero_real: poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 order_poly_real: poly_real > poly_poly_real > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oorder_001t__Real__Oreal,type,
% 0.29/0.47 order_real: real > poly_real > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opcr__poly_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.47 pcr_poly_nat_nat: ( nat > nat > $o ) > ( nat > nat ) > poly_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opcr__poly_001t__Polynomial__Opoly_It__Nat__Onat_J_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.29/0.47 pcr_po273983709ly_nat: ( poly_nat > poly_nat > $o ) > ( nat > poly_nat ) > poly_poly_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opcr__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 pcr_po1200519205y_real: ( poly_poly_real > poly_poly_real > $o ) > ( nat > poly_poly_real ) > poly_poly_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opcr__poly_001t__Polynomial__Opoly_It__Real__Oreal_J_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 pcr_po1314690837y_real: ( poly_real > poly_real > $o ) > ( nat > poly_real ) > poly_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opcr__poly_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.29/0.47 pcr_poly_real_real: ( real > real > $o ) > ( nat > real ) > poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opderiv_001t__Nat__Onat,type,
% 0.29/0.47 pderiv_nat: poly_nat > poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opderiv_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 pderiv_poly_real: poly_poly_real > poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opderiv_001t__Real__Oreal,type,
% 0.29/0.47 pderiv_real: poly_real > poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
% 0.29/0.47 poly_nat2: poly_nat > nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.29/0.47 poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 poly_poly_real2: poly_poly_real > poly_real > poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
% 0.29/0.47 poly_real2: poly_real > real > real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat,type,
% 0.29/0.47 coeff_nat: poly_nat > nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.29/0.47 coeff_poly_nat: poly_poly_nat > nat > poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 coeff_poly_poly_real: poly_poly_poly_real > nat > poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 coeff_poly_real: poly_poly_real > nat > poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Real__Oreal,type,
% 0.29/0.47 coeff_real: poly_real > nat > real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat,type,
% 0.29/0.47 poly_cutoff_nat: nat > poly_nat > poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly__cutoff_001t__Real__Oreal,type,
% 0.29/0.47 poly_cutoff_real: nat > poly_real > poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat,type,
% 0.29/0.47 poly_shift_nat: nat > poly_nat > poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly__shift_001t__Real__Oreal,type,
% 0.29/0.47 poly_shift_real: nat > poly_real > poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
% 0.29/0.47 reflect_poly_nat: poly_nat > poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.29/0.47 reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 reflec1522834046y_real: poly_poly_real > poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal,type,
% 0.29/0.47 reflect_poly_real: poly_real > poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Orsquarefree_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 rsquar1555552848y_real: poly_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Orsquarefree_001t__Real__Oreal,type,
% 0.29/0.47 rsquarefree_real: poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Osynthetic__div_001t__Nat__Onat,type,
% 0.29/0.47 synthetic_div_nat: poly_nat > nat > poly_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Osynthetic__div_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 synthe1498897281y_real: poly_poly_real > poly_real > poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Osynthetic__div_001t__Real__Oreal,type,
% 0.29/0.47 synthetic_div_real: poly_real > real > poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 0.29/0.47 field_1537545994s_real: set_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 0.29/0.47 dvd_dvd_nat: nat > nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.29/0.47 dvd_dvd_poly_nat: poly_nat > poly_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 dvd_dv1946063458y_real: poly_poly_real > poly_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 dvd_dvd_poly_real: poly_real > poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 0.29/0.47 dvd_dvd_real: real > real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.29/0.47 collect_nat: ( nat > $o ) > set_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 collec927113489y_real: ( poly_poly_real > $o ) > set_poly_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 collect_poly_real: ( poly_real > $o ) > set_poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.29/0.47 collect_real: ( real > $o ) > set_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member_001t__Nat__Onat,type,
% 0.29/0.47 member_nat: nat > set_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.29/0.47 member1159720147y_real: poly_poly_real > set_poly_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.29/0.47 member_poly_real: poly_real > set_poly_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member_001t__Real__Oreal,type,
% 0.29/0.47 member_real: real > set_real > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_n____,type,
% 0.29/0.47 n: nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_p____,type,
% 0.29/0.47 p: poly_real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_x,type,
% 0.29/0.47 x: real ).
% 0.29/0.47
% 0.29/0.47 % Relevant facts (344)
% 0.29/0.47 thf(fact_0_poly__roots__finite,axiom,
% 0.29/0.47 ! [P: poly_poly_poly_real] :
% 0.29/0.47 ( ( P != zero_z935034829y_real )
% 0.29/0.47 => ( finite1328464339y_real
% 0.29/0.47 @ ( collec927113489y_real
% 0.29/0.47 @ ^ [X: poly_poly_real] :
% 0.29/0.47 ( ( poly_poly_poly_real2 @ P @ X )
% 0.29/0.47 = zero_z1423781445y_real ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_roots_finite
% 0.29/0.47 thf(fact_1_poly__roots__finite,axiom,
% 0.29/0.47 ! [P: poly_poly_real] :
% 0.29/0.47 ( ( P != zero_z1423781445y_real )
% 0.29/0.47 => ( finite1810960971y_real
% 0.29/0.47 @ ( collect_poly_real
% 0.29/0.47 @ ^ [X: poly_real] :
% 0.29/0.47 ( ( poly_poly_real2 @ P @ X )
% 0.29/0.47 = zero_zero_poly_real ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_roots_finite
% 0.29/0.47 thf(fact_2_poly__roots__finite,axiom,
% 0.29/0.47 ! [P: poly_real] :
% 0.29/0.47 ( ( P != zero_zero_poly_real )
% 0.29/0.47 => ( finite_finite_real
% 0.29/0.47 @ ( collect_real
% 0.29/0.47 @ ^ [X: real] :
% 0.29/0.47 ( ( poly_real2 @ P @ X )
% 0.29/0.47 = zero_zero_real ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_roots_finite
% 0.29/0.47 thf(fact_3_p_I2_J,axiom,
% 0.29/0.47 p != zero_zero_poly_real ).
% 0.29/0.47
% 0.29/0.47 % p(2)
% 0.29/0.47 thf(fact_4__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062,axiom,
% 0.29/0.47 ( ( pderiv_real @ p )
% 0.29/0.47 != zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % \<open>pderiv p \<noteq> 0\<close>
% 0.29/0.47 thf(fact_5_p_I3_J,axiom,
% 0.29/0.47 ( ( poly_real2 @ p @ x )
% 0.29/0.47 = zero_zero_real ) ).
% 0.29/0.47
% 0.29/0.47 % p(3)
% 0.29/0.47 thf(fact_6_poly__0,axiom,
% 0.29/0.47 ! [X2: poly_poly_real] :
% 0.29/0.47 ( ( poly_poly_poly_real2 @ zero_z935034829y_real @ X2 )
% 0.29/0.47 = zero_z1423781445y_real ) ).
% 0.29/0.47
% 0.29/0.47 % poly_0
% 0.29/0.47 thf(fact_7_poly__0,axiom,
% 0.29/0.47 ! [X2: poly_nat] :
% 0.29/0.47 ( ( poly_poly_nat2 @ zero_z1059985641ly_nat @ X2 )
% 0.29/0.47 = zero_zero_poly_nat ) ).
% 0.29/0.47
% 0.29/0.47 % poly_0
% 0.29/0.47 thf(fact_8_poly__0,axiom,
% 0.29/0.47 ! [X2: poly_real] :
% 0.29/0.47 ( ( poly_poly_real2 @ zero_z1423781445y_real @ X2 )
% 0.29/0.47 = zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % poly_0
% 0.29/0.47 thf(fact_9_poly__0,axiom,
% 0.29/0.47 ! [X2: nat] :
% 0.29/0.47 ( ( poly_nat2 @ zero_zero_poly_nat @ X2 )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % poly_0
% 0.29/0.47 thf(fact_10_poly__0,axiom,
% 0.29/0.47 ! [X2: real] :
% 0.29/0.47 ( ( poly_real2 @ zero_zero_poly_real @ X2 )
% 0.29/0.47 = zero_zero_real ) ).
% 0.29/0.47
% 0.29/0.47 % poly_0
% 0.29/0.47 thf(fact_11_finite__Collect__conjI,axiom,
% 0.29/0.47 ! [P2: poly_real > $o,Q: poly_real > $o] :
% 0.29/0.47 ( ( ( finite1810960971y_real @ ( collect_poly_real @ P2 ) )
% 0.29/0.47 | ( finite1810960971y_real @ ( collect_poly_real @ Q ) ) )
% 0.29/0.47 => ( finite1810960971y_real
% 0.29/0.47 @ ( collect_poly_real
% 0.29/0.47 @ ^ [X: poly_real] :
% 0.29/0.47 ( ( P2 @ X )
% 0.29/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_Collect_conjI
% 0.29/0.47 thf(fact_12_finite__Collect__conjI,axiom,
% 0.29/0.47 ! [P2: real > $o,Q: real > $o] :
% 0.29/0.47 ( ( ( finite_finite_real @ ( collect_real @ P2 ) )
% 0.29/0.47 | ( finite_finite_real @ ( collect_real @ Q ) ) )
% 0.29/0.47 => ( finite_finite_real
% 0.29/0.47 @ ( collect_real
% 0.29/0.47 @ ^ [X: real] :
% 0.29/0.47 ( ( P2 @ X )
% 0.29/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_Collect_conjI
% 0.29/0.47 thf(fact_13_finite__Collect__conjI,axiom,
% 0.29/0.47 ! [P2: nat > $o,Q: nat > $o] :
% 0.29/0.47 ( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 0.29/0.47 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 0.29/0.47 => ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [X: nat] :
% 0.29/0.47 ( ( P2 @ X )
% 0.29/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_Collect_conjI
% 0.29/0.47 thf(fact_14_finite__Collect__disjI,axiom,
% 0.29/0.47 ! [P2: poly_real > $o,Q: poly_real > $o] :
% 0.29/0.47 ( ( finite1810960971y_real
% 0.29/0.47 @ ( collect_poly_real
% 0.29/0.47 @ ^ [X: poly_real] :
% 0.29/0.47 ( ( P2 @ X )
% 0.29/0.47 | ( Q @ X ) ) ) )
% 0.29/0.47 = ( ( finite1810960971y_real @ ( collect_poly_real @ P2 ) )
% 0.29/0.47 & ( finite1810960971y_real @ ( collect_poly_real @ Q ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_Collect_disjI
% 0.29/0.47 thf(fact_15_finite__Collect__disjI,axiom,
% 0.29/0.47 ! [P2: real > $o,Q: real > $o] :
% 0.29/0.47 ( ( finite_finite_real
% 0.29/0.47 @ ( collect_real
% 0.29/0.47 @ ^ [X: real] :
% 0.29/0.47 ( ( P2 @ X )
% 0.29/0.47 | ( Q @ X ) ) ) )
% 0.29/0.47 = ( ( finite_finite_real @ ( collect_real @ P2 ) )
% 0.29/0.47 & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_Collect_disjI
% 0.29/0.47 thf(fact_16_finite__Collect__disjI,axiom,
% 0.29/0.47 ! [P2: nat > $o,Q: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [X: nat] :
% 0.29/0.47 ( ( P2 @ X )
% 0.29/0.47 | ( Q @ X ) ) ) )
% 0.29/0.47 = ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 0.29/0.47 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_Collect_disjI
% 0.29/0.47 thf(fact_17_poly__all__0__iff__0,axiom,
% 0.29/0.47 ! [P: poly_poly_poly_real] :
% 0.29/0.47 ( ( ! [X: poly_poly_real] :
% 0.29/0.47 ( ( poly_poly_poly_real2 @ P @ X )
% 0.29/0.47 = zero_z1423781445y_real ) )
% 0.29/0.47 = ( P = zero_z935034829y_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_all_0_iff_0
% 0.29/0.47 thf(fact_18_poly__all__0__iff__0,axiom,
% 0.29/0.47 ! [P: poly_real] :
% 0.29/0.47 ( ( ! [X: real] :
% 0.29/0.47 ( ( poly_real2 @ P @ X )
% 0.29/0.47 = zero_zero_real ) )
% 0.29/0.47 = ( P = zero_zero_poly_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_all_0_iff_0
% 0.29/0.47 thf(fact_19_poly__all__0__iff__0,axiom,
% 0.29/0.47 ! [P: poly_poly_real] :
% 0.29/0.47 ( ( ! [X: poly_real] :
% 0.29/0.47 ( ( poly_poly_real2 @ P @ X )
% 0.29/0.47 = zero_zero_poly_real ) )
% 0.29/0.47 = ( P = zero_z1423781445y_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_all_0_iff_0
% 0.29/0.47 thf(fact_20_rsquarefree__roots,axiom,
% 0.29/0.47 ( rsquarefree_real
% 0.29/0.47 = ( ^ [P3: poly_real] :
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ~ ( ( ( poly_real2 @ P3 @ A )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 & ( ( poly_real2 @ ( pderiv_real @ P3 ) @ A )
% 0.29/0.47 = zero_zero_real ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rsquarefree_roots
% 0.29/0.47 thf(fact_21_not__finite__existsD,axiom,
% 0.29/0.47 ! [P2: real > $o] :
% 0.29/0.47 ( ~ ( finite_finite_real @ ( collect_real @ P2 ) )
% 0.29/0.47 => ? [X_1: real] : ( P2 @ X_1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_finite_existsD
% 0.29/0.47 thf(fact_22_not__finite__existsD,axiom,
% 0.29/0.47 ! [P2: nat > $o] :
% 0.29/0.47 ( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 0.29/0.47 => ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_finite_existsD
% 0.29/0.47 thf(fact_23_not__finite__existsD,axiom,
% 0.29/0.47 ! [P2: poly_real > $o] :
% 0.29/0.47 ( ~ ( finite1810960971y_real @ ( collect_poly_real @ P2 ) )
% 0.29/0.47 => ? [X_1: poly_real] : ( P2 @ X_1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_finite_existsD
% 0.29/0.47 thf(fact_24_pigeonhole__infinite__rel,axiom,
% 0.29/0.47 ! [A2: set_real,B: set_real,R: real > real > $o] :
% 0.29/0.47 ( ~ ( finite_finite_real @ A2 )
% 0.29/0.47 => ( ( finite_finite_real @ B )
% 0.29/0.47 => ( ! [X3: real] :
% 0.29/0.47 ( ( member_real @ X3 @ A2 )
% 0.29/0.47 => ? [Xa: real] :
% 0.29/0.47 ( ( member_real @ Xa @ B )
% 0.29/0.47 & ( R @ X3 @ Xa ) ) )
% 0.29/0.47 => ? [X3: real] :
% 0.29/0.47 ( ( member_real @ X3 @ B )
% 0.29/0.47 & ~ ( finite_finite_real
% 0.29/0.47 @ ( collect_real
% 0.29/0.47 @ ^ [A: real] :
% 0.29/0.47 ( ( member_real @ A @ A2 )
% 0.29/0.47 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pigeonhole_infinite_rel
% 0.29/0.47 thf(fact_25_pigeonhole__infinite__rel,axiom,
% 0.29/0.47 ! [A2: set_real,B: set_nat,R: real > nat > $o] :
% 0.29/0.47 ( ~ ( finite_finite_real @ A2 )
% 0.29/0.47 => ( ( finite_finite_nat @ B )
% 0.29/0.47 => ( ! [X3: real] :
% 0.29/0.47 ( ( member_real @ X3 @ A2 )
% 0.29/0.47 => ? [Xa: nat] :
% 0.29/0.47 ( ( member_nat @ Xa @ B )
% 0.29/0.47 & ( R @ X3 @ Xa ) ) )
% 0.29/0.47 => ? [X3: nat] :
% 0.29/0.47 ( ( member_nat @ X3 @ B )
% 0.29/0.47 & ~ ( finite_finite_real
% 0.29/0.47 @ ( collect_real
% 0.29/0.47 @ ^ [A: real] :
% 0.29/0.47 ( ( member_real @ A @ A2 )
% 0.29/0.47 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pigeonhole_infinite_rel
% 0.29/0.47 thf(fact_26_pigeonhole__infinite__rel,axiom,
% 0.29/0.47 ! [A2: set_real,B: set_poly_real,R: real > poly_real > $o] :
% 0.29/0.47 ( ~ ( finite_finite_real @ A2 )
% 0.29/0.47 => ( ( finite1810960971y_real @ B )
% 0.29/0.47 => ( ! [X3: real] :
% 0.29/0.47 ( ( member_real @ X3 @ A2 )
% 0.29/0.47 => ? [Xa: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ Xa @ B )
% 0.29/0.47 & ( R @ X3 @ Xa ) ) )
% 0.29/0.47 => ? [X3: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ X3 @ B )
% 0.29/0.47 & ~ ( finite_finite_real
% 0.29/0.47 @ ( collect_real
% 0.29/0.47 @ ^ [A: real] :
% 0.29/0.47 ( ( member_real @ A @ A2 )
% 0.29/0.47 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pigeonhole_infinite_rel
% 0.29/0.47 thf(fact_27_pigeonhole__infinite__rel,axiom,
% 0.29/0.47 ! [A2: set_nat,B: set_real,R: nat > real > $o] :
% 0.29/0.47 ( ~ ( finite_finite_nat @ A2 )
% 0.29/0.47 => ( ( finite_finite_real @ B )
% 0.29/0.47 => ( ! [X3: nat] :
% 0.29/0.47 ( ( member_nat @ X3 @ A2 )
% 0.29/0.47 => ? [Xa: real] :
% 0.29/0.47 ( ( member_real @ Xa @ B )
% 0.29/0.47 & ( R @ X3 @ Xa ) ) )
% 0.29/0.47 => ? [X3: real] :
% 0.29/0.47 ( ( member_real @ X3 @ B )
% 0.29/0.47 & ~ ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [A: nat] :
% 0.29/0.47 ( ( member_nat @ A @ A2 )
% 0.29/0.47 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pigeonhole_infinite_rel
% 0.29/0.47 thf(fact_28_pigeonhole__infinite__rel,axiom,
% 0.29/0.47 ! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
% 0.29/0.47 ( ~ ( finite_finite_nat @ A2 )
% 0.29/0.47 => ( ( finite_finite_nat @ B )
% 0.29/0.47 => ( ! [X3: nat] :
% 0.29/0.47 ( ( member_nat @ X3 @ A2 )
% 0.29/0.47 => ? [Xa: nat] :
% 0.29/0.47 ( ( member_nat @ Xa @ B )
% 0.29/0.47 & ( R @ X3 @ Xa ) ) )
% 0.29/0.47 => ? [X3: nat] :
% 0.29/0.47 ( ( member_nat @ X3 @ B )
% 0.29/0.47 & ~ ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [A: nat] :
% 0.29/0.47 ( ( member_nat @ A @ A2 )
% 0.29/0.47 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pigeonhole_infinite_rel
% 0.29/0.47 thf(fact_29_pigeonhole__infinite__rel,axiom,
% 0.29/0.47 ! [A2: set_nat,B: set_poly_real,R: nat > poly_real > $o] :
% 0.29/0.47 ( ~ ( finite_finite_nat @ A2 )
% 0.29/0.47 => ( ( finite1810960971y_real @ B )
% 0.29/0.47 => ( ! [X3: nat] :
% 0.29/0.47 ( ( member_nat @ X3 @ A2 )
% 0.29/0.47 => ? [Xa: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ Xa @ B )
% 0.29/0.47 & ( R @ X3 @ Xa ) ) )
% 0.29/0.47 => ? [X3: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ X3 @ B )
% 0.29/0.47 & ~ ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [A: nat] :
% 0.29/0.47 ( ( member_nat @ A @ A2 )
% 0.29/0.47 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pigeonhole_infinite_rel
% 0.29/0.47 thf(fact_30_pigeonhole__infinite__rel,axiom,
% 0.29/0.47 ! [A2: set_poly_real,B: set_real,R: poly_real > real > $o] :
% 0.29/0.47 ( ~ ( finite1810960971y_real @ A2 )
% 0.29/0.47 => ( ( finite_finite_real @ B )
% 0.29/0.47 => ( ! [X3: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ X3 @ A2 )
% 0.29/0.47 => ? [Xa: real] :
% 0.29/0.47 ( ( member_real @ Xa @ B )
% 0.29/0.47 & ( R @ X3 @ Xa ) ) )
% 0.29/0.47 => ? [X3: real] :
% 0.29/0.47 ( ( member_real @ X3 @ B )
% 0.29/0.47 & ~ ( finite1810960971y_real
% 0.29/0.47 @ ( collect_poly_real
% 0.29/0.47 @ ^ [A: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ A @ A2 )
% 0.29/0.47 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pigeonhole_infinite_rel
% 0.29/0.47 thf(fact_31_pigeonhole__infinite__rel,axiom,
% 0.29/0.47 ! [A2: set_poly_real,B: set_nat,R: poly_real > nat > $o] :
% 0.29/0.47 ( ~ ( finite1810960971y_real @ A2 )
% 0.29/0.47 => ( ( finite_finite_nat @ B )
% 0.29/0.47 => ( ! [X3: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ X3 @ A2 )
% 0.29/0.47 => ? [Xa: nat] :
% 0.29/0.47 ( ( member_nat @ Xa @ B )
% 0.29/0.47 & ( R @ X3 @ Xa ) ) )
% 0.29/0.47 => ? [X3: nat] :
% 0.29/0.47 ( ( member_nat @ X3 @ B )
% 0.29/0.47 & ~ ( finite1810960971y_real
% 0.29/0.47 @ ( collect_poly_real
% 0.29/0.47 @ ^ [A: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ A @ A2 )
% 0.29/0.47 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pigeonhole_infinite_rel
% 0.29/0.47 thf(fact_32_pigeonhole__infinite__rel,axiom,
% 0.29/0.47 ! [A2: set_poly_real,B: set_poly_real,R: poly_real > poly_real > $o] :
% 0.29/0.47 ( ~ ( finite1810960971y_real @ A2 )
% 0.29/0.47 => ( ( finite1810960971y_real @ B )
% 0.29/0.47 => ( ! [X3: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ X3 @ A2 )
% 0.29/0.47 => ? [Xa: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ Xa @ B )
% 0.29/0.47 & ( R @ X3 @ Xa ) ) )
% 0.29/0.47 => ? [X3: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ X3 @ B )
% 0.29/0.47 & ~ ( finite1810960971y_real
% 0.29/0.47 @ ( collect_poly_real
% 0.29/0.47 @ ^ [A: poly_real] :
% 0.29/0.47 ( ( member_poly_real @ A @ A2 )
% 0.29/0.47 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pigeonhole_infinite_rel
% 0.29/0.47 thf(fact_33_poly__eq__poly__eq__iff,axiom,
% 0.29/0.47 ! [P: poly_real,Q2: poly_real] :
% 0.29/0.47 ( ( ( poly_real2 @ P )
% 0.29/0.47 = ( poly_real2 @ Q2 ) )
% 0.29/0.47 = ( P = Q2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_eq_poly_eq_iff
% 0.29/0.47 thf(fact_34_poly__eq__poly__eq__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_real,Q2: poly_poly_real] :
% 0.29/0.47 ( ( ( poly_poly_real2 @ P )
% 0.29/0.47 = ( poly_poly_real2 @ Q2 ) )
% 0.29/0.47 = ( P = Q2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_eq_poly_eq_iff
% 0.29/0.47 thf(fact_35_n__def,axiom,
% 0.29/0.47 ( n
% 0.29/0.47 = ( degree_real @ p ) ) ).
% 0.29/0.47
% 0.29/0.47 % n_def
% 0.29/0.47 thf(fact_36_assms_I2_J,axiom,
% 0.29/0.47 algebraic_real @ x ).
% 0.29/0.47
% 0.29/0.47 % assms(2)
% 0.29/0.47 thf(fact_37_irrationsl,axiom,
% 0.29/0.47 ~ ( member_real @ x @ field_1537545994s_real ) ).
% 0.29/0.47
% 0.29/0.47 % irrationsl
% 0.29/0.47 thf(fact_38_degree__0,axiom,
% 0.29/0.47 ( ( degree_real @ zero_zero_poly_real )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % degree_0
% 0.29/0.47 thf(fact_39_degree__0,axiom,
% 0.29/0.47 ( ( degree_poly_real @ zero_z1423781445y_real )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % degree_0
% 0.29/0.47 thf(fact_40_degree__0,axiom,
% 0.29/0.47 ( ( degree_nat @ zero_zero_poly_nat )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % degree_0
% 0.29/0.47 thf(fact_41_pderiv__0,axiom,
% 0.29/0.47 ( ( pderiv_real @ zero_zero_poly_real )
% 0.29/0.47 = zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % pderiv_0
% 0.29/0.47 thf(fact_42_pderiv__0,axiom,
% 0.29/0.47 ( ( pderiv_poly_real @ zero_z1423781445y_real )
% 0.29/0.47 = zero_z1423781445y_real ) ).
% 0.29/0.47
% 0.29/0.47 % pderiv_0
% 0.29/0.47 thf(fact_43_pderiv__0,axiom,
% 0.29/0.47 ( ( pderiv_nat @ zero_zero_poly_nat )
% 0.29/0.47 = zero_zero_poly_nat ) ).
% 0.29/0.47
% 0.29/0.47 % pderiv_0
% 0.29/0.47 thf(fact_44_pderiv__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_real] :
% 0.29/0.47 ( ( ( pderiv_real @ P )
% 0.29/0.47 = zero_zero_poly_real )
% 0.29/0.47 = ( ( degree_real @ P )
% 0.29/0.47 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % pderiv_eq_0_iff
% 0.29/0.47 thf(fact_45_pderiv__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_real] :
% 0.29/0.47 ( ( ( pderiv_poly_real @ P )
% 0.29/0.47 = zero_z1423781445y_real )
% 0.29/0.47 = ( ( degree_poly_real @ P )
% 0.29/0.47 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % pderiv_eq_0_iff
% 0.29/0.47 thf(fact_46_pderiv__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_nat] :
% 0.29/0.47 ( ( ( pderiv_nat @ P )
% 0.29/0.47 = zero_zero_poly_nat )
% 0.29/0.47 = ( ( degree_nat @ P )
% 0.29/0.47 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % pderiv_eq_0_iff
% 0.29/0.47 thf(fact_47_zero__reorient,axiom,
% 0.29/0.47 ! [X2: real] :
% 0.29/0.47 ( ( zero_zero_real = X2 )
% 0.29/0.47 = ( X2 = zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_48_zero__reorient,axiom,
% 0.29/0.47 ! [X2: poly_real] :
% 0.29/0.47 ( ( zero_zero_poly_real = X2 )
% 0.29/0.47 = ( X2 = zero_zero_poly_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_49_zero__reorient,axiom,
% 0.29/0.47 ! [X2: nat] :
% 0.29/0.47 ( ( zero_zero_nat = X2 )
% 0.29/0.47 = ( X2 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_50_zero__reorient,axiom,
% 0.29/0.47 ! [X2: poly_poly_real] :
% 0.29/0.47 ( ( zero_z1423781445y_real = X2 )
% 0.29/0.47 = ( X2 = zero_z1423781445y_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_51_zero__reorient,axiom,
% 0.29/0.47 ! [X2: poly_nat] :
% 0.29/0.47 ( ( zero_zero_poly_nat = X2 )
% 0.29/0.47 = ( X2 = zero_zero_poly_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_52_is__zero__null,axiom,
% 0.29/0.47 ( is_zero_real
% 0.29/0.47 = ( ^ [P3: poly_real] : ( P3 = zero_zero_poly_real ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % is_zero_null
% 0.29/0.47 thf(fact_53_is__zero__null,axiom,
% 0.29/0.47 ( is_zero_poly_real
% 0.29/0.47 = ( ^ [P3: poly_poly_real] : ( P3 = zero_z1423781445y_real ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % is_zero_null
% 0.29/0.47 thf(fact_54_is__zero__null,axiom,
% 0.29/0.47 ( is_zero_nat
% 0.29/0.47 = ( ^ [P3: poly_nat] : ( P3 = zero_zero_poly_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % is_zero_null
% 0.29/0.47 thf(fact_55_poly__cutoff__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( poly_cutoff_real @ N @ zero_zero_poly_real )
% 0.29/0.47 = zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % poly_cutoff_0
% 0.29/0.47 thf(fact_56_poly__cutoff__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( poly_c1404107022y_real @ N @ zero_z1423781445y_real )
% 0.29/0.47 = zero_z1423781445y_real ) ).
% 0.29/0.47
% 0.29/0.47 % poly_cutoff_0
% 0.29/0.47 thf(fact_57_poly__cutoff__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
% 0.29/0.47 = zero_zero_poly_nat ) ).
% 0.29/0.47
% 0.29/0.47 % poly_cutoff_0
% 0.29/0.47 thf(fact_58_reflect__poly__at__0__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_real] :
% 0.29/0.47 ( ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 = ( P = zero_zero_poly_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_at_0_eq_0_iff
% 0.29/0.47 thf(fact_59_reflect__poly__at__0__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_real] :
% 0.29/0.47 ( ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
% 0.29/0.47 = zero_zero_poly_real )
% 0.29/0.47 = ( P = zero_z1423781445y_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_at_0_eq_0_iff
% 0.29/0.47 thf(fact_60_reflect__poly__at__0__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_nat] :
% 0.29/0.47 ( ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( P = zero_zero_poly_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_at_0_eq_0_iff
% 0.29/0.47 thf(fact_61_reflect__poly__at__0__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_poly_real] :
% 0.29/0.47 ( ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
% 0.29/0.47 = zero_z1423781445y_real )
% 0.29/0.47 = ( P = zero_z935034829y_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_at_0_eq_0_iff
% 0.29/0.47 thf(fact_62_reflect__poly__at__0__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_nat] :
% 0.29/0.47 ( ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
% 0.29/0.47 = zero_zero_poly_nat )
% 0.29/0.47 = ( P = zero_z1059985641ly_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_at_0_eq_0_iff
% 0.29/0.47 thf(fact_63__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062,axiom,
% 0.29/0.47 ~ ! [P4: poly_real] :
% 0.29/0.47 ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
% 0.29/0.47 => ( ( P4 != zero_zero_poly_real )
% 0.29/0.47 => ( ( poly_real2 @ P4 @ x )
% 0.29/0.47 != zero_zero_real ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % \<open>\<And>thesisa. (\<And>p. \<lbrakk>\<And>i. coeff p i \<in> \<int>; p \<noteq> 0; poly p x = 0\<rbrakk> \<Longrightarrow> thesisa) \<Longrightarrow> thesisa\<close>
% 0.29/0.47 thf(fact_64_synthetic__div__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_real,C: real] :
% 0.29/0.47 ( ( ( synthetic_div_real @ P @ C )
% 0.29/0.47 = zero_zero_poly_real )
% 0.29/0.47 = ( ( degree_real @ P )
% 0.29/0.47 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % synthetic_div_eq_0_iff
% 0.29/0.47 thf(fact_65_synthetic__div__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_real,C: poly_real] :
% 0.29/0.47 ( ( ( synthe1498897281y_real @ P @ C )
% 0.29/0.47 = zero_z1423781445y_real )
% 0.29/0.47 = ( ( degree_poly_real @ P )
% 0.29/0.47 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % synthetic_div_eq_0_iff
% 0.29/0.47 thf(fact_66_synthetic__div__eq__0__iff,axiom,
% 0.29/0.47 ! [P: poly_nat,C: nat] :
% 0.29/0.47 ( ( ( synthetic_div_nat @ P @ C )
% 0.29/0.47 = zero_zero_poly_nat )
% 0.29/0.47 = ( ( degree_nat @ P )
% 0.29/0.47 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % synthetic_div_eq_0_iff
% 0.29/0.47 thf(fact_67_poly__shift__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( poly_shift_real @ N @ zero_zero_poly_real )
% 0.29/0.47 = zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % poly_shift_0
% 0.29/0.47 thf(fact_68_poly__shift__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( poly_shift_poly_real @ N @ zero_z1423781445y_real )
% 0.29/0.47 = zero_z1423781445y_real ) ).
% 0.29/0.47
% 0.29/0.47 % poly_shift_0
% 0.29/0.47 thf(fact_69_poly__shift__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( poly_shift_nat @ N @ zero_zero_poly_nat )
% 0.29/0.47 = zero_zero_poly_nat ) ).
% 0.29/0.47
% 0.29/0.47 % poly_shift_0
% 0.29/0.47 thf(fact_70_order__root,axiom,
% 0.29/0.47 ! [P: poly_real,A3: real] :
% 0.29/0.47 ( ( ( poly_real2 @ P @ A3 )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 = ( ( P = zero_zero_poly_real )
% 0.29/0.47 | ( ( order_real @ A3 @ P )
% 0.29/0.47 != zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_root
% 0.29/0.47 thf(fact_71_order__root,axiom,
% 0.29/0.47 ! [P: poly_poly_real,A3: poly_real] :
% 0.29/0.47 ( ( ( poly_poly_real2 @ P @ A3 )
% 0.29/0.47 = zero_zero_poly_real )
% 0.29/0.47 = ( ( P = zero_z1423781445y_real )
% 0.29/0.47 | ( ( order_poly_real @ A3 @ P )
% 0.29/0.47 != zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_root
% 0.29/0.47 thf(fact_72_order__root,axiom,
% 0.29/0.47 ! [P: poly_poly_poly_real,A3: poly_poly_real] :
% 0.29/0.47 ( ( ( poly_poly_poly_real2 @ P @ A3 )
% 0.29/0.47 = zero_z1423781445y_real )
% 0.29/0.47 = ( ( P = zero_z935034829y_real )
% 0.29/0.47 | ( ( order_poly_poly_real @ A3 @ P )
% 0.29/0.47 != zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_root
% 0.29/0.47 thf(fact_73_leading__coeff__0__iff,axiom,
% 0.29/0.47 ! [P: poly_real] :
% 0.29/0.47 ( ( ( coeff_real @ P @ ( degree_real @ P ) )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 = ( P = zero_zero_poly_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % leading_coeff_0_iff
% 0.29/0.47 thf(fact_74_leading__coeff__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_real] :
% 0.29/0.47 ( ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
% 0.29/0.47 = zero_zero_poly_real )
% 0.29/0.47 = ( P = zero_z1423781445y_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % leading_coeff_0_iff
% 0.29/0.47 thf(fact_75_leading__coeff__0__iff,axiom,
% 0.29/0.47 ! [P: poly_nat] :
% 0.29/0.47 ( ( ( coeff_nat @ P @ ( degree_nat @ P ) )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( P = zero_zero_poly_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % leading_coeff_0_iff
% 0.29/0.47 thf(fact_76_leading__coeff__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_poly_real] :
% 0.29/0.47 ( ( ( coeff_poly_poly_real @ P @ ( degree360860553y_real @ P ) )
% 0.29/0.47 = zero_z1423781445y_real )
% 0.29/0.47 = ( P = zero_z935034829y_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % leading_coeff_0_iff
% 0.29/0.47 thf(fact_77_leading__coeff__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_nat] :
% 0.29/0.47 ( ( ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) )
% 0.29/0.47 = zero_zero_poly_nat )
% 0.29/0.47 = ( P = zero_z1059985641ly_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % leading_coeff_0_iff
% 0.29/0.47 thf(fact_78_divide__poly__main__0,axiom,
% 0.29/0.47 ! [R2: poly_real,D: poly_real,Dr: nat,N: nat] :
% 0.29/0.47 ( ( divide1561404011n_real @ zero_zero_real @ zero_zero_poly_real @ R2 @ D @ Dr @ N )
% 0.29/0.47 = zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % divide_poly_main_0
% 0.29/0.47 thf(fact_79_divide__poly__main__0,axiom,
% 0.29/0.47 ! [R2: poly_poly_real,D: poly_poly_real,Dr: nat,N: nat] :
% 0.29/0.47 ( ( divide1142363123y_real @ zero_zero_poly_real @ zero_z1423781445y_real @ R2 @ D @ Dr @ N )
% 0.29/0.47 = zero_z1423781445y_real ) ).
% 0.29/0.47
% 0.29/0.47 % divide_poly_main_0
% 0.29/0.47 thf(fact_80_divide__poly__main__0,axiom,
% 0.29/0.47 ! [R2: poly_poly_poly_real,D: poly_poly_poly_real,Dr: nat,N: nat] :
% 0.29/0.47 ( ( divide924636027y_real @ zero_z1423781445y_real @ zero_z935034829y_real @ R2 @ D @ Dr @ N )
% 0.29/0.47 = zero_z935034829y_real ) ).
% 0.29/0.47
% 0.29/0.47 % divide_poly_main_0
% 0.29/0.47 thf(fact_81_zero__poly_Otransfer,axiom,
% 0.29/0.47 ( pcr_poly_real_real
% 0.29/0.47 @ ^ [Y: real,Z: real] : ( Y = Z )
% 0.29/0.47 @ ^ [Uu: nat] : zero_zero_real
% 0.29/0.47 @ zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % zero_poly.transfer
% 0.29/0.47 thf(fact_82_zero__poly_Otransfer,axiom,
% 0.29/0.47 ( pcr_po1314690837y_real
% 0.29/0.47 @ ^ [Y: poly_real,Z: poly_real] : ( Y = Z )
% 0.29/0.47 @ ^ [Uu: nat] : zero_zero_poly_real
% 0.29/0.47 @ zero_z1423781445y_real ) ).
% 0.29/0.47
% 0.29/0.47 % zero_poly.transfer
% 0.29/0.47 thf(fact_83_zero__poly_Otransfer,axiom,
% 0.29/0.47 ( pcr_poly_nat_nat
% 0.29/0.47 @ ^ [Y: nat,Z: nat] : ( Y = Z )
% 0.29/0.47 @ ^ [Uu: nat] : zero_zero_nat
% 0.29/0.47 @ zero_zero_poly_nat ) ).
% 0.29/0.47
% 0.29/0.47 % zero_poly.transfer
% 0.29/0.47 thf(fact_84_zero__poly_Otransfer,axiom,
% 0.29/0.47 ( pcr_po1200519205y_real
% 0.29/0.47 @ ^ [Y: poly_poly_real,Z: poly_poly_real] : ( Y = Z )
% 0.29/0.47 @ ^ [Uu: nat] : zero_z1423781445y_real
% 0.29/0.47 @ zero_z935034829y_real ) ).
% 0.29/0.47
% 0.29/0.47 % zero_poly.transfer
% 0.29/0.47 thf(fact_85_zero__poly_Otransfer,axiom,
% 0.29/0.47 ( pcr_po273983709ly_nat
% 0.29/0.47 @ ^ [Y: poly_nat,Z: poly_nat] : ( Y = Z )
% 0.29/0.47 @ ^ [Uu: nat] : zero_zero_poly_nat
% 0.29/0.47 @ zero_z1059985641ly_nat ) ).
% 0.29/0.47
% 0.29/0.47 % zero_poly.transfer
% 0.29/0.47 thf(fact_86_p_I1_J,axiom,
% 0.29/0.47 ! [I2: nat] : ( member_real @ ( coeff_real @ p @ I2 ) @ ring_1_Ints_real ) ).
% 0.29/0.47
% 0.29/0.47 % p(1)
% 0.29/0.47 thf(fact_87_reflect__poly__0,axiom,
% 0.29/0.47 ( ( reflect_poly_real @ zero_zero_poly_real )
% 0.29/0.47 = zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_0
% 0.29/0.47 thf(fact_88_reflect__poly__0,axiom,
% 0.29/0.47 ( ( reflec1522834046y_real @ zero_z1423781445y_real )
% 0.29/0.47 = zero_z1423781445y_real ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_0
% 0.29/0.47 thf(fact_89_reflect__poly__0,axiom,
% 0.29/0.47 ( ( reflect_poly_nat @ zero_zero_poly_nat )
% 0.29/0.47 = zero_zero_poly_nat ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_0
% 0.29/0.47 thf(fact_90_synthetic__div__0,axiom,
% 0.29/0.47 ! [C: real] :
% 0.29/0.47 ( ( synthetic_div_real @ zero_zero_poly_real @ C )
% 0.29/0.47 = zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % synthetic_div_0
% 0.29/0.47 thf(fact_91_synthetic__div__0,axiom,
% 0.29/0.47 ! [C: poly_real] :
% 0.29/0.47 ( ( synthe1498897281y_real @ zero_z1423781445y_real @ C )
% 0.29/0.47 = zero_z1423781445y_real ) ).
% 0.29/0.47
% 0.29/0.47 % synthetic_div_0
% 0.29/0.47 thf(fact_92_synthetic__div__0,axiom,
% 0.29/0.47 ! [C: nat] :
% 0.29/0.47 ( ( synthetic_div_nat @ zero_zero_poly_nat @ C )
% 0.29/0.47 = zero_zero_poly_nat ) ).
% 0.29/0.47
% 0.29/0.47 % synthetic_div_0
% 0.29/0.47 thf(fact_93_coeff__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( coeff_poly_poly_real @ zero_z935034829y_real @ N )
% 0.29/0.47 = zero_z1423781445y_real ) ).
% 0.29/0.47
% 0.29/0.47 % coeff_0
% 0.29/0.47 thf(fact_94_coeff__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( coeff_poly_nat @ zero_z1059985641ly_nat @ N )
% 0.29/0.47 = zero_zero_poly_nat ) ).
% 0.29/0.47
% 0.29/0.47 % coeff_0
% 0.29/0.47 thf(fact_95_coeff__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( coeff_real @ zero_zero_poly_real @ N )
% 0.29/0.47 = zero_zero_real ) ).
% 0.29/0.47
% 0.29/0.47 % coeff_0
% 0.29/0.47 thf(fact_96_coeff__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( coeff_poly_real @ zero_z1423781445y_real @ N )
% 0.29/0.47 = zero_zero_poly_real ) ).
% 0.29/0.47
% 0.29/0.47 % coeff_0
% 0.29/0.47 thf(fact_97_coeff__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( coeff_nat @ zero_zero_poly_nat @ N )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % coeff_0
% 0.29/0.47 thf(fact_98_reflect__poly__reflect__poly,axiom,
% 0.29/0.47 ! [P: poly_real] :
% 0.29/0.47 ( ( ( coeff_real @ P @ zero_zero_nat )
% 0.29/0.47 != zero_zero_real )
% 0.29/0.47 => ( ( reflect_poly_real @ ( reflect_poly_real @ P ) )
% 0.29/0.47 = P ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_reflect_poly
% 0.29/0.47 thf(fact_99_reflect__poly__reflect__poly,axiom,
% 0.29/0.47 ! [P: poly_poly_real] :
% 0.29/0.47 ( ( ( coeff_poly_real @ P @ zero_zero_nat )
% 0.29/0.47 != zero_zero_poly_real )
% 0.29/0.47 => ( ( reflec1522834046y_real @ ( reflec1522834046y_real @ P ) )
% 0.29/0.47 = P ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_reflect_poly
% 0.29/0.47 thf(fact_100_reflect__poly__reflect__poly,axiom,
% 0.29/0.47 ! [P: poly_nat] :
% 0.29/0.47 ( ( ( coeff_nat @ P @ zero_zero_nat )
% 0.29/0.47 != zero_zero_nat )
% 0.29/0.47 => ( ( reflect_poly_nat @ ( reflect_poly_nat @ P ) )
% 0.29/0.47 = P ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_reflect_poly
% 0.29/0.47 thf(fact_101_reflect__poly__reflect__poly,axiom,
% 0.29/0.47 ! [P: poly_poly_poly_real] :
% 0.29/0.47 ( ( ( coeff_poly_poly_real @ P @ zero_zero_nat )
% 0.29/0.47 != zero_z1423781445y_real )
% 0.29/0.47 => ( ( reflec144234502y_real @ ( reflec144234502y_real @ P ) )
% 0.29/0.47 = P ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_reflect_poly
% 0.29/0.47 thf(fact_102_reflect__poly__reflect__poly,axiom,
% 0.29/0.47 ! [P: poly_poly_nat] :
% 0.29/0.47 ( ( ( coeff_poly_nat @ P @ zero_zero_nat )
% 0.29/0.47 != zero_zero_poly_nat )
% 0.29/0.47 => ( ( reflec781175074ly_nat @ ( reflec781175074ly_nat @ P ) )
% 0.29/0.47 = P ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_reflect_poly
% 0.29/0.47 thf(fact_103_coeff__0__reflect__poly,axiom,
% 0.29/0.47 ! [P: poly_real] :
% 0.29/0.47 ( ( coeff_real @ ( reflect_poly_real @ P ) @ zero_zero_nat )
% 0.29/0.47 = ( coeff_real @ P @ ( degree_real @ P ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % coeff_0_reflect_poly
% 0.29/0.47 thf(fact_104_coeff__0__reflect__poly__0__iff,axiom,
% 0.29/0.47 ! [P: poly_real] :
% 0.29/0.47 ( ( ( coeff_real @ ( reflect_poly_real @ P ) @ zero_zero_nat )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 = ( P = zero_zero_poly_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % coeff_0_reflect_poly_0_iff
% 0.29/0.47 thf(fact_105_coeff__0__reflect__poly__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_real] :
% 0.29/0.47 ( ( ( coeff_poly_real @ ( reflec1522834046y_real @ P ) @ zero_zero_nat )
% 0.29/0.47 = zero_zero_poly_real )
% 0.29/0.47 = ( P = zero_z1423781445y_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % coeff_0_reflect_poly_0_iff
% 0.29/0.47 thf(fact_106_coeff__0__reflect__poly__0__iff,axiom,
% 0.29/0.47 ! [P: poly_nat] :
% 0.29/0.47 ( ( ( coeff_nat @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( P = zero_zero_poly_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % coeff_0_reflect_poly_0_iff
% 0.29/0.47 thf(fact_107_coeff__0__reflect__poly__0__iff,axiom,
% 0.29/0.47 ! [P: poly_poly_poly_real] :
% 0.29/0.48 ( ( ( coeff_poly_poly_real @ ( reflec144234502y_real @ P ) @ zero_zero_nat )
% 0.29/0.48 = zero_z1423781445y_real )
% 0.29/0.48 = ( P = zero_z935034829y_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % coeff_0_reflect_poly_0_iff
% 0.29/0.48 thf(fact_108_coeff__0__reflect__poly__0__iff,axiom,
% 0.29/0.48 ! [P: poly_poly_nat] :
% 0.29/0.48 ( ( ( coeff_poly_nat @ ( reflec781175074ly_nat @ P ) @ zero_zero_nat )
% 0.29/0.48 = zero_zero_poly_nat )
% 0.29/0.48 = ( P = zero_z1059985641ly_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % coeff_0_reflect_poly_0_iff
% 0.29/0.48 thf(fact_109_degree__reflect__poly__eq,axiom,
% 0.29/0.48 ! [P: poly_real] :
% 0.29/0.48 ( ( ( coeff_real @ P @ zero_zero_nat )
% 0.29/0.48 != zero_zero_real )
% 0.29/0.48 => ( ( degree_real @ ( reflect_poly_real @ P ) )
% 0.29/0.48 = ( degree_real @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % degree_reflect_poly_eq
% 0.29/0.48 thf(fact_110_degree__reflect__poly__eq,axiom,
% 0.29/0.48 ! [P: poly_poly_real] :
% 0.29/0.48 ( ( ( coeff_poly_real @ P @ zero_zero_nat )
% 0.29/0.48 != zero_zero_poly_real )
% 0.29/0.48 => ( ( degree_poly_real @ ( reflec1522834046y_real @ P ) )
% 0.29/0.48 = ( degree_poly_real @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % degree_reflect_poly_eq
% 0.29/0.48 thf(fact_111_degree__reflect__poly__eq,axiom,
% 0.29/0.48 ! [P: poly_nat] :
% 0.29/0.48 ( ( ( coeff_nat @ P @ zero_zero_nat )
% 0.29/0.48 != zero_zero_nat )
% 0.29/0.48 => ( ( degree_nat @ ( reflect_poly_nat @ P ) )
% 0.29/0.48 = ( degree_nat @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % degree_reflect_poly_eq
% 0.29/0.48 thf(fact_112_degree__reflect__poly__eq,axiom,
% 0.29/0.48 ! [P: poly_poly_poly_real] :
% 0.29/0.48 ( ( ( coeff_poly_poly_real @ P @ zero_zero_nat )
% 0.29/0.48 != zero_z1423781445y_real )
% 0.29/0.48 => ( ( degree360860553y_real @ ( reflec144234502y_real @ P ) )
% 0.29/0.48 = ( degree360860553y_real @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % degree_reflect_poly_eq
% 0.29/0.48 thf(fact_113_degree__reflect__poly__eq,axiom,
% 0.29/0.48 ! [P: poly_poly_nat] :
% 0.29/0.48 ( ( ( coeff_poly_nat @ P @ zero_zero_nat )
% 0.29/0.48 != zero_zero_poly_nat )
% 0.29/0.48 => ( ( degree_poly_nat @ ( reflec781175074ly_nat @ P ) )
% 0.29/0.48 = ( degree_poly_nat @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % degree_reflect_poly_eq
% 0.29/0.48 thf(fact_114_poly__reflect__poly__0,axiom,
% 0.29/0.48 ! [P: poly_real] :
% 0.29/0.48 ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
% 0.29/0.48 = ( coeff_real @ P @ ( degree_real @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_reflect_poly_0
% 0.29/0.48 thf(fact_115_poly__reflect__poly__0,axiom,
% 0.29/0.48 ! [P: poly_poly_real] :
% 0.29/0.48 ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
% 0.29/0.48 = ( coeff_poly_real @ P @ ( degree_poly_real @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_reflect_poly_0
% 0.29/0.48 thf(fact_116_poly__reflect__poly__0,axiom,
% 0.29/0.48 ! [P: poly_nat] :
% 0.29/0.48 ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 0.29/0.48 = ( coeff_nat @ P @ ( degree_nat @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_reflect_poly_0
% 0.29/0.48 thf(fact_117_poly__reflect__poly__0,axiom,
% 0.29/0.48 ! [P: poly_poly_poly_real] :
% 0.29/0.48 ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
% 0.29/0.48 = ( coeff_poly_poly_real @ P @ ( degree360860553y_real @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_reflect_poly_0
% 0.29/0.48 thf(fact_118_poly__reflect__poly__0,axiom,
% 0.29/0.48 ! [P: poly_poly_nat] :
% 0.29/0.48 ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
% 0.29/0.48 = ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_reflect_poly_0
% 0.29/0.48 thf(fact_119_mem__Collect__eq,axiom,
% 0.29/0.48 ! [A3: real,P2: real > $o] :
% 0.29/0.48 ( ( member_real @ A3 @ ( collect_real @ P2 ) )
% 0.29/0.48 = ( P2 @ A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % mem_Collect_eq
% 0.29/0.48 thf(fact_120_mem__Collect__eq,axiom,
% 0.29/0.48 ! [A3: nat,P2: nat > $o] :
% 0.29/0.48 ( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
% 0.29/0.48 = ( P2 @ A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % mem_Collect_eq
% 0.29/0.48 thf(fact_121_mem__Collect__eq,axiom,
% 0.29/0.48 ! [A3: poly_real,P2: poly_real > $o] :
% 0.29/0.48 ( ( member_poly_real @ A3 @ ( collect_poly_real @ P2 ) )
% 0.29/0.48 = ( P2 @ A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % mem_Collect_eq
% 0.29/0.48 thf(fact_122_Collect__mem__eq,axiom,
% 0.29/0.48 ! [A2: set_real] :
% 0.29/0.48 ( ( collect_real
% 0.29/0.48 @ ^ [X: real] : ( member_real @ X @ A2 ) )
% 0.29/0.48 = A2 ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_mem_eq
% 0.29/0.48 thf(fact_123_Collect__mem__eq,axiom,
% 0.29/0.48 ! [A2: set_nat] :
% 0.29/0.48 ( ( collect_nat
% 0.29/0.48 @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
% 0.29/0.48 = A2 ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_mem_eq
% 0.29/0.48 thf(fact_124_Collect__mem__eq,axiom,
% 0.29/0.48 ! [A2: set_poly_real] :
% 0.29/0.48 ( ( collect_poly_real
% 0.29/0.48 @ ^ [X: poly_real] : ( member_poly_real @ X @ A2 ) )
% 0.29/0.48 = A2 ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_mem_eq
% 0.29/0.48 thf(fact_125_Collect__cong,axiom,
% 0.29/0.48 ! [P2: real > $o,Q: real > $o] :
% 0.29/0.48 ( ! [X3: real] :
% 0.29/0.48 ( ( P2 @ X3 )
% 0.29/0.48 = ( Q @ X3 ) )
% 0.29/0.48 => ( ( collect_real @ P2 )
% 0.29/0.48 = ( collect_real @ Q ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_cong
% 0.29/0.48 thf(fact_126_Collect__cong,axiom,
% 0.29/0.48 ! [P2: nat > $o,Q: nat > $o] :
% 0.29/0.48 ( ! [X3: nat] :
% 0.29/0.48 ( ( P2 @ X3 )
% 0.29/0.48 = ( Q @ X3 ) )
% 0.29/0.48 => ( ( collect_nat @ P2 )
% 0.29/0.48 = ( collect_nat @ Q ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_cong
% 0.29/0.48 thf(fact_127_Collect__cong,axiom,
% 0.29/0.48 ! [P2: poly_real > $o,Q: poly_real > $o] :
% 0.29/0.48 ( ! [X3: poly_real] :
% 0.29/0.48 ( ( P2 @ X3 )
% 0.29/0.48 = ( Q @ X3 ) )
% 0.29/0.48 => ( ( collect_poly_real @ P2 )
% 0.29/0.48 = ( collect_poly_real @ Q ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_cong
% 0.29/0.48 thf(fact_128_poly__eqI,axiom,
% 0.29/0.48 ! [P: poly_real,Q2: poly_real] :
% 0.29/0.48 ( ! [N2: nat] :
% 0.29/0.48 ( ( coeff_real @ P @ N2 )
% 0.29/0.48 = ( coeff_real @ Q2 @ N2 ) )
% 0.29/0.48 => ( P = Q2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_eqI
% 0.29/0.48 thf(fact_129_poly__eq__iff,axiom,
% 0.29/0.48 ( ( ^ [Y: poly_real,Z: poly_real] : ( Y = Z ) )
% 0.29/0.48 = ( ^ [P3: poly_real,Q3: poly_real] :
% 0.29/0.48 ! [N3: nat] :
% 0.29/0.48 ( ( coeff_real @ P3 @ N3 )
% 0.29/0.48 = ( coeff_real @ Q3 @ N3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_eq_iff
% 0.29/0.48 thf(fact_130_coeff__inject,axiom,
% 0.29/0.48 ! [X2: poly_real,Y2: poly_real] :
% 0.29/0.48 ( ( ( coeff_real @ X2 )
% 0.29/0.48 = ( coeff_real @ Y2 ) )
% 0.29/0.48 = ( X2 = Y2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % coeff_inject
% 0.29/0.48 thf(fact_131_algebraicE,axiom,
% 0.29/0.48 ! [X2: real] :
% 0.29/0.48 ( ( algebraic_real @ X2 )
% 0.29/0.48 => ~ ! [P4: poly_real] :
% 0.29/0.48 ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
% 0.29/0.48 => ( ( P4 != zero_zero_poly_real )
% 0.29/0.48 => ( ( poly_real2 @ P4 @ X2 )
% 0.29/0.48 != zero_zero_real ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraicE
% 0.29/0.48 thf(fact_132_algebraicI,axiom,
% 0.29/0.48 ! [P: poly_real,X2: real] :
% 0.29/0.48 ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
% 0.29/0.48 => ( ( P != zero_zero_poly_real )
% 0.29/0.48 => ( ( ( poly_real2 @ P @ X2 )
% 0.29/0.48 = zero_zero_real )
% 0.29/0.48 => ( algebraic_real @ X2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraicI
% 0.29/0.48 thf(fact_133_algebraic__def,axiom,
% 0.29/0.48 ( algebraic_real
% 0.29/0.48 = ( ^ [X: real] :
% 0.29/0.48 ? [P3: poly_real] :
% 0.29/0.48 ( ! [I4: nat] : ( member_real @ ( coeff_real @ P3 @ I4 ) @ ring_1_Ints_real )
% 0.29/0.48 & ( P3 != zero_zero_poly_real )
% 0.29/0.48 & ( ( poly_real2 @ P3 @ X )
% 0.29/0.48 = zero_zero_real ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraic_def
% 0.29/0.48 thf(fact_134_algebraic__altdef,axiom,
% 0.29/0.48 ( algebraic_real
% 0.29/0.48 = ( ^ [X: real] :
% 0.29/0.48 ? [P3: poly_real] :
% 0.29/0.48 ( ! [I4: nat] : ( member_real @ ( coeff_real @ P3 @ I4 ) @ field_1537545994s_real )
% 0.29/0.48 & ( P3 != zero_zero_poly_real )
% 0.29/0.48 & ( ( poly_real2 @ P3 @ X )
% 0.29/0.48 = zero_zero_real ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraic_altdef
% 0.29/0.48 thf(fact_135_zero__poly_Orep__eq,axiom,
% 0.29/0.48 ( ( coeff_poly_poly_real @ zero_z935034829y_real )
% 0.29/0.48 = ( ^ [Uu: nat] : zero_z1423781445y_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % zero_poly.rep_eq
% 0.29/0.48 thf(fact_136_zero__poly_Orep__eq,axiom,
% 0.29/0.48 ( ( coeff_poly_nat @ zero_z1059985641ly_nat )
% 0.29/0.48 = ( ^ [Uu: nat] : zero_zero_poly_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % zero_poly.rep_eq
% 0.29/0.48 thf(fact_137_zero__poly_Orep__eq,axiom,
% 0.29/0.48 ( ( coeff_real @ zero_zero_poly_real )
% 0.29/0.48 = ( ^ [Uu: nat] : zero_zero_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % zero_poly.rep_eq
% 0.29/0.48 thf(fact_138_zero__poly_Orep__eq,axiom,
% 0.29/0.48 ( ( coeff_poly_real @ zero_z1423781445y_real )
% 0.29/0.48 = ( ^ [Uu: nat] : zero_zero_poly_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % zero_poly.rep_eq
% 0.29/0.48 thf(fact_139_zero__poly_Orep__eq,axiom,
% 0.29/0.48 ( ( coeff_nat @ zero_zero_poly_nat )
% 0.29/0.48 = ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % zero_poly.rep_eq
% 0.29/0.48 thf(fact_140_poly__0__coeff__0,axiom,
% 0.29/0.48 ! [P: poly_real] :
% 0.29/0.48 ( ( poly_real2 @ P @ zero_zero_real )
% 0.29/0.48 = ( coeff_real @ P @ zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_0_coeff_0
% 0.29/0.48 thf(fact_141_poly__0__coeff__0,axiom,
% 0.29/0.48 ! [P: poly_poly_real] :
% 0.29/0.48 ( ( poly_poly_real2 @ P @ zero_zero_poly_real )
% 0.29/0.48 = ( coeff_poly_real @ P @ zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_0_coeff_0
% 0.29/0.48 thf(fact_142_poly__0__coeff__0,axiom,
% 0.29/0.48 ! [P: poly_nat] :
% 0.29/0.48 ( ( poly_nat2 @ P @ zero_zero_nat )
% 0.29/0.48 = ( coeff_nat @ P @ zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_0_coeff_0
% 0.29/0.48 thf(fact_143_poly__0__coeff__0,axiom,
% 0.29/0.48 ! [P: poly_poly_poly_real] :
% 0.29/0.48 ( ( poly_poly_poly_real2 @ P @ zero_z1423781445y_real )
% 0.29/0.48 = ( coeff_poly_poly_real @ P @ zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_0_coeff_0
% 0.29/0.48 thf(fact_144_poly__0__coeff__0,axiom,
% 0.29/0.48 ! [P: poly_poly_nat] :
% 0.29/0.48 ( ( poly_poly_nat2 @ P @ zero_zero_poly_nat )
% 0.29/0.48 = ( coeff_poly_nat @ P @ zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_0_coeff_0
% 0.29/0.48 thf(fact_145_order__0I,axiom,
% 0.29/0.48 ! [P: poly_real,A3: real] :
% 0.29/0.48 ( ( ( poly_real2 @ P @ A3 )
% 0.29/0.48 != zero_zero_real )
% 0.29/0.48 => ( ( order_real @ A3 @ P )
% 0.29/0.48 = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_0I
% 0.29/0.48 thf(fact_146_order__0I,axiom,
% 0.29/0.48 ! [P: poly_poly_real,A3: poly_real] :
% 0.29/0.48 ( ( ( poly_poly_real2 @ P @ A3 )
% 0.29/0.48 != zero_zero_poly_real )
% 0.29/0.48 => ( ( order_poly_real @ A3 @ P )
% 0.29/0.48 = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_0I
% 0.29/0.48 thf(fact_147_order__0I,axiom,
% 0.29/0.48 ! [P: poly_poly_poly_real,A3: poly_poly_real] :
% 0.29/0.48 ( ( ( poly_poly_poly_real2 @ P @ A3 )
% 0.29/0.48 != zero_z1423781445y_real )
% 0.29/0.48 => ( ( order_poly_poly_real @ A3 @ P )
% 0.29/0.48 = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_0I
% 0.29/0.48 thf(fact_148_leading__coeff__neq__0,axiom,
% 0.29/0.48 ! [P: poly_poly_poly_real] :
% 0.29/0.48 ( ( P != zero_z935034829y_real )
% 0.29/0.48 => ( ( coeff_poly_poly_real @ P @ ( degree360860553y_real @ P ) )
% 0.29/0.48 != zero_z1423781445y_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % leading_coeff_neq_0
% 0.29/0.48 thf(fact_149_leading__coeff__neq__0,axiom,
% 0.29/0.48 ! [P: poly_poly_nat] :
% 0.29/0.48 ( ( P != zero_z1059985641ly_nat )
% 0.29/0.48 => ( ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) )
% 0.29/0.48 != zero_zero_poly_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % leading_coeff_neq_0
% 0.29/0.48 thf(fact_150_leading__coeff__neq__0,axiom,
% 0.29/0.48 ! [P: poly_real] :
% 0.29/0.48 ( ( P != zero_zero_poly_real )
% 0.29/0.48 => ( ( coeff_real @ P @ ( degree_real @ P ) )
% 0.29/0.48 != zero_zero_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % leading_coeff_neq_0
% 0.29/0.48 thf(fact_151_leading__coeff__neq__0,axiom,
% 0.29/0.48 ! [P: poly_poly_real] :
% 0.29/0.48 ( ( P != zero_z1423781445y_real )
% 0.29/0.48 => ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
% 0.29/0.48 != zero_zero_poly_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % leading_coeff_neq_0
% 0.29/0.48 thf(fact_152_leading__coeff__neq__0,axiom,
% 0.29/0.48 ! [P: poly_nat] :
% 0.29/0.48 ( ( P != zero_zero_poly_nat )
% 0.29/0.48 => ( ( coeff_nat @ P @ ( degree_nat @ P ) )
% 0.29/0.48 != zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % leading_coeff_neq_0
% 0.29/0.48 thf(fact_153_Rats__infinite,axiom,
% 0.29/0.48 ~ ( finite_finite_real @ field_1537545994s_real ) ).
% 0.29/0.48
% 0.29/0.48 % Rats_infinite
% 0.29/0.48 thf(fact_154_Rats__0,axiom,
% 0.29/0.48 member_real @ zero_zero_real @ field_1537545994s_real ).
% 0.29/0.48
% 0.29/0.48 % Rats_0
% 0.29/0.48 thf(fact_155_Ints__0,axiom,
% 0.29/0.48 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 0.29/0.48
% 0.29/0.48 % Ints_0
% 0.29/0.48 thf(fact_156_Ints__0,axiom,
% 0.29/0.48 member_poly_real @ zero_zero_poly_real @ ring_1690226883y_real ).
% 0.29/0.48
% 0.29/0.48 % Ints_0
% 0.29/0.48 thf(fact_157_Ints__0,axiom,
% 0.29/0.48 member1159720147y_real @ zero_z1423781445y_real @ ring_1897377867y_real ).
% 0.29/0.48
% 0.29/0.48 % Ints_0
% 0.29/0.48 thf(fact_158_rsquarefree__def,axiom,
% 0.29/0.48 ( rsquarefree_real
% 0.29/0.48 = ( ^ [P3: poly_real] :
% 0.29/0.48 ( ( P3 != zero_zero_poly_real )
% 0.29/0.48 & ! [A: real] :
% 0.29/0.48 ( ( ( order_real @ A @ P3 )
% 0.29/0.48 = zero_zero_nat )
% 0.29/0.48 | ( ( order_real @ A @ P3 )
% 0.29/0.48 = one_one_nat ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % rsquarefree_def
% 0.29/0.48 thf(fact_159_rsquarefree__def,axiom,
% 0.29/0.48 ( rsquar1555552848y_real
% 0.29/0.48 = ( ^ [P3: poly_poly_real] :
% 0.29/0.48 ( ( P3 != zero_z1423781445y_real )
% 0.29/0.48 & ! [A: poly_real] :
% 0.29/0.48 ( ( ( order_poly_real @ A @ P3 )
% 0.29/0.48 = zero_zero_nat )
% 0.29/0.48 | ( ( order_poly_real @ A @ P3 )
% 0.29/0.48 = one_one_nat ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % rsquarefree_def
% 0.29/0.48 thf(fact_160_poly__cutoff__1,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ( N = zero_zero_nat )
% 0.29/0.48 => ( ( poly_cutoff_real @ N @ one_one_poly_real )
% 0.29/0.48 = zero_zero_poly_real ) )
% 0.29/0.48 & ( ( N != zero_zero_nat )
% 0.29/0.48 => ( ( poly_cutoff_real @ N @ one_one_poly_real )
% 0.29/0.48 = one_one_poly_real ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_cutoff_1
% 0.29/0.48 thf(fact_161_poly__cutoff__1,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ( N = zero_zero_nat )
% 0.29/0.48 => ( ( poly_c1404107022y_real @ N @ one_on501200385y_real )
% 0.29/0.48 = zero_z1423781445y_real ) )
% 0.29/0.48 & ( ( N != zero_zero_nat )
% 0.29/0.48 => ( ( poly_c1404107022y_real @ N @ one_on501200385y_real )
% 0.29/0.48 = one_on501200385y_real ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_cutoff_1
% 0.29/0.48 thf(fact_162_poly__cutoff__1,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ( N = zero_zero_nat )
% 0.29/0.48 => ( ( poly_cutoff_nat @ N @ one_one_poly_nat )
% 0.29/0.48 = zero_zero_poly_nat ) )
% 0.29/0.48 & ( ( N != zero_zero_nat )
% 0.29/0.48 => ( ( poly_cutoff_nat @ N @ one_one_poly_nat )
% 0.29/0.48 = one_one_poly_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_cutoff_1
% 0.29/0.48 thf(fact_163_order__pderiv2,axiom,
% 0.29/0.48 ! [P: poly_real,A3: real,N: nat] :
% 0.29/0.48 ( ( ( pderiv_real @ P )
% 0.29/0.48 != zero_zero_poly_real )
% 0.29/0.48 => ( ( ( order_real @ A3 @ P )
% 0.29/0.48 != zero_zero_nat )
% 0.29/0.48 => ( ( ( order_real @ A3 @ ( pderiv_real @ P ) )
% 0.29/0.48 = N )
% 0.29/0.48 = ( ( order_real @ A3 @ P )
% 0.29/0.48 = ( suc @ N ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_pderiv2
% 0.29/0.48 thf(fact_164_order__pderiv,axiom,
% 0.29/0.48 ! [P: poly_real,A3: real] :
% 0.29/0.48 ( ( ( pderiv_real @ P )
% 0.29/0.48 != zero_zero_poly_real )
% 0.29/0.48 => ( ( ( order_real @ A3 @ P )
% 0.29/0.48 != zero_zero_nat )
% 0.29/0.48 => ( ( order_real @ A3 @ P )
% 0.29/0.48 = ( suc @ ( order_real @ A3 @ ( pderiv_real @ P ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_pderiv
% 0.29/0.48 thf(fact_165_cr__poly__def,axiom,
% 0.29/0.48 ( cr_poly_real
% 0.29/0.48 = ( ^ [X: nat > real,Y3: poly_real] :
% 0.29/0.48 ( X
% 0.29/0.48 = ( coeff_real @ Y3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % cr_poly_def
% 0.29/0.48 thf(fact_166_dvd__pderiv__iff,axiom,
% 0.29/0.48 ! [P: poly_real] :
% 0.29/0.48 ( ( dvd_dvd_poly_real @ P @ ( pderiv_real @ P ) )
% 0.29/0.48 = ( ( degree_real @ P )
% 0.29/0.48 = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_pderiv_iff
% 0.29/0.48 thf(fact_167_degree__1,axiom,
% 0.29/0.48 ( ( degree_real @ one_one_poly_real )
% 0.29/0.48 = zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % degree_1
% 0.29/0.48 thf(fact_168_poly__1,axiom,
% 0.29/0.48 ! [X2: real] :
% 0.29/0.48 ( ( poly_real2 @ one_one_poly_real @ X2 )
% 0.29/0.48 = one_one_real ) ).
% 0.29/0.48
% 0.29/0.48 % poly_1
% 0.29/0.48 thf(fact_169_poly__1,axiom,
% 0.29/0.48 ! [X2: poly_real] :
% 0.29/0.48 ( ( poly_poly_real2 @ one_on501200385y_real @ X2 )
% 0.29/0.48 = one_one_poly_real ) ).
% 0.29/0.48
% 0.29/0.48 % poly_1
% 0.29/0.48 thf(fact_170_poly__1,axiom,
% 0.29/0.48 ! [X2: nat] :
% 0.29/0.48 ( ( poly_nat2 @ one_one_poly_nat @ X2 )
% 0.29/0.48 = one_one_nat ) ).
% 0.29/0.48
% 0.29/0.48 % poly_1
% 0.29/0.48 thf(fact_171_pderiv__1,axiom,
% 0.29/0.48 ( ( pderiv_real @ one_one_poly_real )
% 0.29/0.48 = zero_zero_poly_real ) ).
% 0.29/0.48
% 0.29/0.48 % pderiv_1
% 0.29/0.48 thf(fact_172_pderiv__1,axiom,
% 0.29/0.48 ( ( pderiv_poly_real @ one_on501200385y_real )
% 0.29/0.48 = zero_z1423781445y_real ) ).
% 0.29/0.48
% 0.29/0.48 % pderiv_1
% 0.29/0.48 thf(fact_173_pderiv__1,axiom,
% 0.29/0.48 ( ( pderiv_nat @ one_one_poly_nat )
% 0.29/0.48 = zero_zero_poly_nat ) ).
% 0.29/0.48
% 0.29/0.48 % pderiv_1
% 0.29/0.48 thf(fact_174_lead__coeff__1,axiom,
% 0.29/0.48 ( ( coeff_nat @ one_one_poly_nat @ ( degree_nat @ one_one_poly_nat ) )
% 0.29/0.48 = one_one_nat ) ).
% 0.29/0.48
% 0.29/0.48 % lead_coeff_1
% 0.29/0.48 thf(fact_175_lead__coeff__1,axiom,
% 0.29/0.48 ( ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) )
% 0.29/0.48 = one_one_real ) ).
% 0.29/0.48
% 0.29/0.48 % lead_coeff_1
% 0.29/0.48 thf(fact_176_one__reorient,axiom,
% 0.29/0.48 ! [X2: nat] :
% 0.29/0.48 ( ( one_one_nat = X2 )
% 0.29/0.48 = ( X2 = one_one_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % one_reorient
% 0.29/0.48 thf(fact_177_Ints__1,axiom,
% 0.29/0.48 member_real @ one_one_real @ ring_1_Ints_real ).
% 0.29/0.48
% 0.29/0.48 % Ints_1
% 0.29/0.48 thf(fact_178_Rats__1,axiom,
% 0.29/0.48 member_real @ one_one_real @ field_1537545994s_real ).
% 0.29/0.48
% 0.29/0.48 % Rats_1
% 0.29/0.48 thf(fact_179_is__unit__iff__degree,axiom,
% 0.29/0.48 ! [P: poly_real] :
% 0.29/0.48 ( ( P != zero_zero_poly_real )
% 0.29/0.48 => ( ( dvd_dvd_poly_real @ P @ one_one_poly_real )
% 0.29/0.48 = ( ( degree_real @ P )
% 0.29/0.48 = zero_zero_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % is_unit_iff_degree
% 0.29/0.48 thf(fact_180_not__dvd__pderiv,axiom,
% 0.29/0.48 ! [P: poly_real] :
% 0.29/0.48 ( ( ( degree_real @ P )
% 0.29/0.48 != zero_zero_nat )
% 0.29/0.48 => ~ ( dvd_dvd_poly_real @ P @ ( pderiv_real @ P ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % not_dvd_pderiv
% 0.29/0.48 thf(fact_181_poly__shift__1,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ( N = zero_zero_nat )
% 0.29/0.48 => ( ( poly_shift_real @ N @ one_one_poly_real )
% 0.29/0.48 = one_one_poly_real ) )
% 0.29/0.48 & ( ( N != zero_zero_nat )
% 0.29/0.48 => ( ( poly_shift_real @ N @ one_one_poly_real )
% 0.29/0.48 = zero_zero_poly_real ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_shift_1
% 0.29/0.48 thf(fact_182_poly__shift__1,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ( N = zero_zero_nat )
% 0.29/0.48 => ( ( poly_shift_poly_real @ N @ one_on501200385y_real )
% 0.29/0.48 = one_on501200385y_real ) )
% 0.29/0.48 & ( ( N != zero_zero_nat )
% 0.29/0.48 => ( ( poly_shift_poly_real @ N @ one_on501200385y_real )
% 0.29/0.48 = zero_z1423781445y_real ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_shift_1
% 0.29/0.48 thf(fact_183_poly__shift__1,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ( N = zero_zero_nat )
% 0.29/0.48 => ( ( poly_shift_nat @ N @ one_one_poly_nat )
% 0.29/0.48 = one_one_poly_nat ) )
% 0.29/0.48 & ( ( N != zero_zero_nat )
% 0.29/0.48 => ( ( poly_shift_nat @ N @ one_one_poly_nat )
% 0.29/0.48 = zero_zero_poly_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_shift_1
% 0.29/0.48 thf(fact_184_dvd__0__left__iff,axiom,
% 0.29/0.48 ! [A3: real] :
% 0.29/0.48 ( ( dvd_dvd_real @ zero_zero_real @ A3 )
% 0.29/0.48 = ( A3 = zero_zero_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left_iff
% 0.29/0.48 thf(fact_185_dvd__0__left__iff,axiom,
% 0.29/0.48 ! [A3: poly_real] :
% 0.29/0.48 ( ( dvd_dvd_poly_real @ zero_zero_poly_real @ A3 )
% 0.29/0.48 = ( A3 = zero_zero_poly_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left_iff
% 0.29/0.48 thf(fact_186_dvd__0__left__iff,axiom,
% 0.29/0.48 ! [A3: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 0.29/0.48 = ( A3 = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left_iff
% 0.29/0.48 thf(fact_187_dvd__0__left__iff,axiom,
% 0.29/0.48 ! [A3: poly_poly_real] :
% 0.29/0.48 ( ( dvd_dv1946063458y_real @ zero_z1423781445y_real @ A3 )
% 0.29/0.48 = ( A3 = zero_z1423781445y_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left_iff
% 0.29/0.48 thf(fact_188_dvd__0__left__iff,axiom,
% 0.29/0.48 ! [A3: poly_nat] :
% 0.29/0.48 ( ( dvd_dvd_poly_nat @ zero_zero_poly_nat @ A3 )
% 0.29/0.48 = ( A3 = zero_zero_poly_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left_iff
% 0.29/0.48 thf(fact_189_dvd__0__right,axiom,
% 0.29/0.48 ! [A3: real] : ( dvd_dvd_real @ A3 @ zero_zero_real ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_right
% 0.29/0.48 thf(fact_190_dvd__0__right,axiom,
% 0.29/0.48 ! [A3: poly_real] : ( dvd_dvd_poly_real @ A3 @ zero_zero_poly_real ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_right
% 0.29/0.48 thf(fact_191_dvd__0__right,axiom,
% 0.29/0.48 ! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_right
% 0.29/0.48 thf(fact_192_dvd__0__right,axiom,
% 0.29/0.48 ! [A3: poly_poly_real] : ( dvd_dv1946063458y_real @ A3 @ zero_z1423781445y_real ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_right
% 0.29/0.48 thf(fact_193_dvd__0__right,axiom,
% 0.29/0.48 ! [A3: poly_nat] : ( dvd_dvd_poly_nat @ A3 @ zero_zero_poly_nat ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_right
% 0.29/0.48 thf(fact_194_One__nat__def,axiom,
% 0.29/0.48 ( one_one_nat
% 0.29/0.48 = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % One_nat_def
% 0.29/0.48 thf(fact_195_not__is__unit__0,axiom,
% 0.29/0.48 ~ ( dvd_dvd_poly_real @ zero_zero_poly_real @ one_one_poly_real ) ).
% 0.29/0.48
% 0.29/0.48 % not_is_unit_0
% 0.29/0.48 thf(fact_196_not__is__unit__0,axiom,
% 0.29/0.48 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.29/0.48
% 0.29/0.48 % not_is_unit_0
% 0.29/0.48 thf(fact_197_not__is__unit__0,axiom,
% 0.29/0.48 ~ ( dvd_dv1946063458y_real @ zero_z1423781445y_real @ one_on501200385y_real ) ).
% 0.29/0.48
% 0.29/0.48 % not_is_unit_0
% 0.29/0.48 thf(fact_198_nat_Oinject,axiom,
% 0.29/0.48 ! [X22: nat,Y22: nat] :
% 0.29/0.48 ( ( ( suc @ X22 )
% 0.29/0.48 = ( suc @ Y22 ) )
% 0.29/0.48 = ( X22 = Y22 ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat.inject
% 0.29/0.48 thf(fact_199_old_Onat_Oinject,axiom,
% 0.29/0.48 ! [Nat: nat,Nat2: nat] :
% 0.29/0.48 ( ( ( suc @ Nat )
% 0.29/0.48 = ( suc @ Nat2 ) )
% 0.29/0.48 = ( Nat = Nat2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % old.nat.inject
% 0.29/0.48 thf(fact_200_nat__dvd__1__iff__1,axiom,
% 0.29/0.48 ! [M: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.29/0.48 = ( M = one_one_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat_dvd_1_iff_1
% 0.29/0.48 thf(fact_201_dvd__1__iff__1,axiom,
% 0.29/0.48 ! [M: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.29/0.48 = ( M
% 0.29/0.48 = ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_1_iff_1
% 0.29/0.48 thf(fact_202_dvd__1__left,axiom,
% 0.29/0.48 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_1_left
% 0.29/0.48 thf(fact_203_dvd__trans,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat,C: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 => ( ( dvd_dvd_nat @ B2 @ C )
% 0.29/0.48 => ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_trans
% 0.29/0.48 thf(fact_204_dvd__refl,axiom,
% 0.29/0.48 ! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_refl
% 0.29/0.48 thf(fact_205_n__not__Suc__n,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( N
% 0.29/0.48 != ( suc @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % n_not_Suc_n
% 0.29/0.48 thf(fact_206_Suc__inject,axiom,
% 0.29/0.48 ! [X2: nat,Y2: nat] :
% 0.29/0.48 ( ( ( suc @ X2 )
% 0.29/0.48 = ( suc @ Y2 ) )
% 0.29/0.48 => ( X2 = Y2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % Suc_inject
% 0.29/0.48 thf(fact_207_zero__neq__one,axiom,
% 0.29/0.48 zero_zero_real != one_one_real ).
% 0.29/0.48
% 0.29/0.48 % zero_neq_one
% 0.29/0.48 thf(fact_208_zero__neq__one,axiom,
% 0.29/0.48 zero_zero_poly_real != one_one_poly_real ).
% 0.29/0.48
% 0.29/0.48 % zero_neq_one
% 0.29/0.48 thf(fact_209_zero__neq__one,axiom,
% 0.29/0.48 zero_zero_nat != one_one_nat ).
% 0.29/0.48
% 0.29/0.48 % zero_neq_one
% 0.29/0.48 thf(fact_210_zero__neq__one,axiom,
% 0.29/0.48 zero_z1423781445y_real != one_on501200385y_real ).
% 0.29/0.48
% 0.29/0.48 % zero_neq_one
% 0.29/0.48 thf(fact_211_zero__neq__one,axiom,
% 0.29/0.48 zero_zero_poly_nat != one_one_poly_nat ).
% 0.29/0.48
% 0.29/0.48 % zero_neq_one
% 0.29/0.48 thf(fact_212_dvd__0__left,axiom,
% 0.29/0.48 ! [A3: real] :
% 0.29/0.48 ( ( dvd_dvd_real @ zero_zero_real @ A3 )
% 0.29/0.48 => ( A3 = zero_zero_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left
% 0.29/0.48 thf(fact_213_dvd__0__left,axiom,
% 0.29/0.48 ! [A3: poly_real] :
% 0.29/0.48 ( ( dvd_dvd_poly_real @ zero_zero_poly_real @ A3 )
% 0.29/0.48 => ( A3 = zero_zero_poly_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left
% 0.29/0.48 thf(fact_214_dvd__0__left,axiom,
% 0.29/0.48 ! [A3: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 0.29/0.48 => ( A3 = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left
% 0.29/0.48 thf(fact_215_dvd__0__left,axiom,
% 0.29/0.48 ! [A3: poly_poly_real] :
% 0.29/0.48 ( ( dvd_dv1946063458y_real @ zero_z1423781445y_real @ A3 )
% 0.29/0.48 => ( A3 = zero_z1423781445y_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left
% 0.29/0.48 thf(fact_216_dvd__0__left,axiom,
% 0.29/0.48 ! [A3: poly_nat] :
% 0.29/0.48 ( ( dvd_dvd_poly_nat @ zero_zero_poly_nat @ A3 )
% 0.29/0.48 => ( A3 = zero_zero_poly_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_0_left
% 0.29/0.48 thf(fact_217_dvd__unit__imp__unit,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.29/0.48 => ( dvd_dvd_nat @ A3 @ one_one_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_unit_imp_unit
% 0.29/0.48 thf(fact_218_unit__imp__dvd,axiom,
% 0.29/0.48 ! [B2: nat,A3: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.29/0.48 => ( dvd_dvd_nat @ B2 @ A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % unit_imp_dvd
% 0.29/0.48 thf(fact_219_one__dvd,axiom,
% 0.29/0.48 ! [A3: nat] : ( dvd_dvd_nat @ one_one_nat @ A3 ) ).
% 0.29/0.48
% 0.29/0.48 % one_dvd
% 0.29/0.48 thf(fact_220_not0__implies__Suc,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( N != zero_zero_nat )
% 0.29/0.48 => ? [M2: nat] :
% 0.29/0.48 ( N
% 0.29/0.48 = ( suc @ M2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % not0_implies_Suc
% 0.29/0.48 thf(fact_221_old_Onat_Oinducts,axiom,
% 0.29/0.48 ! [P2: nat > $o,Nat: nat] :
% 0.29/0.48 ( ( P2 @ zero_zero_nat )
% 0.29/0.48 => ( ! [Nat3: nat] :
% 0.29/0.48 ( ( P2 @ Nat3 )
% 0.29/0.48 => ( P2 @ ( suc @ Nat3 ) ) )
% 0.29/0.48 => ( P2 @ Nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % old.nat.inducts
% 0.29/0.48 thf(fact_222_old_Onat_Oexhaust,axiom,
% 0.29/0.48 ! [Y2: nat] :
% 0.29/0.48 ( ( Y2 != zero_zero_nat )
% 0.29/0.48 => ~ ! [Nat3: nat] :
% 0.29/0.48 ( Y2
% 0.29/0.48 != ( suc @ Nat3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % old.nat.exhaust
% 0.29/0.48 thf(fact_223_Zero__not__Suc,axiom,
% 0.29/0.48 ! [M: nat] :
% 0.29/0.48 ( zero_zero_nat
% 0.29/0.48 != ( suc @ M ) ) ).
% 0.29/0.48
% 0.29/0.48 % Zero_not_Suc
% 0.29/0.48 thf(fact_224_Zero__neq__Suc,axiom,
% 0.29/0.48 ! [M: nat] :
% 0.29/0.48 ( zero_zero_nat
% 0.29/0.48 != ( suc @ M ) ) ).
% 0.29/0.48
% 0.29/0.48 % Zero_neq_Suc
% 0.29/0.48 thf(fact_225_Suc__neq__Zero,axiom,
% 0.29/0.48 ! [M: nat] :
% 0.29/0.48 ( ( suc @ M )
% 0.29/0.48 != zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % Suc_neq_Zero
% 0.29/0.48 thf(fact_226_zero__induct,axiom,
% 0.29/0.48 ! [P2: nat > $o,K: nat] :
% 0.29/0.48 ( ( P2 @ K )
% 0.29/0.48 => ( ! [N2: nat] :
% 0.29/0.48 ( ( P2 @ ( suc @ N2 ) )
% 0.29/0.48 => ( P2 @ N2 ) )
% 0.29/0.48 => ( P2 @ zero_zero_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % zero_induct
% 0.29/0.48 thf(fact_227_diff__induct,axiom,
% 0.29/0.48 ! [P2: nat > nat > $o,M: nat,N: nat] :
% 0.29/0.48 ( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
% 0.29/0.48 => ( ! [Y4: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y4 ) )
% 0.29/0.48 => ( ! [X3: nat,Y4: nat] :
% 0.29/0.48 ( ( P2 @ X3 @ Y4 )
% 0.29/0.48 => ( P2 @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 0.29/0.48 => ( P2 @ M @ N ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % diff_induct
% 0.29/0.48 thf(fact_228_nat__induct,axiom,
% 0.29/0.48 ! [P2: nat > $o,N: nat] :
% 0.29/0.48 ( ( P2 @ zero_zero_nat )
% 0.29/0.48 => ( ! [N2: nat] :
% 0.29/0.48 ( ( P2 @ N2 )
% 0.29/0.48 => ( P2 @ ( suc @ N2 ) ) )
% 0.29/0.48 => ( P2 @ N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat_induct
% 0.29/0.48 thf(fact_229_nat_OdiscI,axiom,
% 0.29/0.48 ! [Nat: nat,X22: nat] :
% 0.29/0.48 ( ( Nat
% 0.29/0.48 = ( suc @ X22 ) )
% 0.29/0.48 => ( Nat != zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat.discI
% 0.29/0.48 thf(fact_230_old_Onat_Odistinct_I1_J,axiom,
% 0.29/0.48 ! [Nat2: nat] :
% 0.29/0.48 ( zero_zero_nat
% 0.29/0.48 != ( suc @ Nat2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % old.nat.distinct(1)
% 0.29/0.48 thf(fact_231_old_Onat_Odistinct_I2_J,axiom,
% 0.29/0.48 ! [Nat2: nat] :
% 0.29/0.48 ( ( suc @ Nat2 )
% 0.29/0.48 != zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % old.nat.distinct(2)
% 0.29/0.48 thf(fact_232_nat_Odistinct_I1_J,axiom,
% 0.29/0.48 ! [X22: nat] :
% 0.29/0.48 ( zero_zero_nat
% 0.29/0.48 != ( suc @ X22 ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat.distinct(1)
% 0.29/0.48 thf(fact_233_algebraic__int_Oinducts,axiom,
% 0.29/0.48 ! [X2: real,P2: real > $o] :
% 0.29/0.48 ( ( algebraic_int_real @ X2 )
% 0.29/0.48 => ( ! [P4: poly_real,X3: real] :
% 0.29/0.48 ( ( ( coeff_real @ P4 @ ( degree_real @ P4 ) )
% 0.29/0.48 = one_one_real )
% 0.29/0.48 => ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
% 0.29/0.48 => ( ( ( poly_real2 @ P4 @ X3 )
% 0.29/0.48 = zero_zero_real )
% 0.29/0.48 => ( P2 @ X3 ) ) ) )
% 0.29/0.48 => ( P2 @ X2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraic_int.inducts
% 0.29/0.48 thf(fact_234_algebraic__int_Ointros,axiom,
% 0.29/0.48 ! [P: poly_real,X2: real] :
% 0.29/0.48 ( ( ( coeff_real @ P @ ( degree_real @ P ) )
% 0.29/0.48 = one_one_real )
% 0.29/0.48 => ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
% 0.29/0.48 => ( ( ( poly_real2 @ P @ X2 )
% 0.29/0.48 = zero_zero_real )
% 0.29/0.48 => ( algebraic_int_real @ X2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraic_int.intros
% 0.29/0.48 thf(fact_235_algebraic__int_Osimps,axiom,
% 0.29/0.48 ( algebraic_int_real
% 0.29/0.48 = ( ^ [A: real] :
% 0.29/0.48 ? [P3: poly_real,X: real] :
% 0.29/0.48 ( ( A = X )
% 0.29/0.48 & ( ( coeff_real @ P3 @ ( degree_real @ P3 ) )
% 0.29/0.48 = one_one_real )
% 0.29/0.48 & ! [I4: nat] : ( member_real @ ( coeff_real @ P3 @ I4 ) @ ring_1_Ints_real )
% 0.29/0.48 & ( ( poly_real2 @ P3 @ X )
% 0.29/0.48 = zero_zero_real ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraic_int.simps
% 0.29/0.48 thf(fact_236_algebraic__int_Ocases,axiom,
% 0.29/0.48 ! [A3: real] :
% 0.29/0.48 ( ( algebraic_int_real @ A3 )
% 0.29/0.48 => ~ ! [P4: poly_real] :
% 0.29/0.48 ( ( ( coeff_real @ P4 @ ( degree_real @ P4 ) )
% 0.29/0.48 = one_one_real )
% 0.29/0.48 => ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
% 0.29/0.48 => ( ( poly_real2 @ P4 @ A3 )
% 0.29/0.48 != zero_zero_real ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraic_int.cases
% 0.29/0.48 thf(fact_237_exists__least__lemma,axiom,
% 0.29/0.48 ! [P2: nat > $o] :
% 0.29/0.48 ( ~ ( P2 @ zero_zero_nat )
% 0.29/0.48 => ( ? [X_12: nat] : ( P2 @ X_12 )
% 0.29/0.48 => ? [N2: nat] :
% 0.29/0.48 ( ~ ( P2 @ N2 )
% 0.29/0.48 & ( P2 @ ( suc @ N2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % exists_least_lemma
% 0.29/0.48 thf(fact_238_algebraic__int__0,axiom,
% 0.29/0.48 algebraic_int_real @ zero_zero_real ).
% 0.29/0.48
% 0.29/0.48 % algebraic_int_0
% 0.29/0.48 thf(fact_239_dvd__antisym,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ M @ N )
% 0.29/0.48 => ( ( dvd_dvd_nat @ N @ M )
% 0.29/0.48 => ( M = N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_antisym
% 0.29/0.48 thf(fact_240_int__imp__algebraic__int,axiom,
% 0.29/0.48 ! [X2: real] :
% 0.29/0.48 ( ( member_real @ X2 @ ring_1_Ints_real )
% 0.29/0.48 => ( algebraic_int_real @ X2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % int_imp_algebraic_int
% 0.29/0.48 thf(fact_241_algebraic__int__imp__algebraic,axiom,
% 0.29/0.48 ! [X2: real] :
% 0.29/0.48 ( ( algebraic_int_real @ X2 )
% 0.29/0.48 => ( algebraic_real @ X2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraic_int_imp_algebraic
% 0.29/0.48 thf(fact_242_rational__algebraic__int__is__int,axiom,
% 0.29/0.48 ! [X2: real] :
% 0.29/0.48 ( ( algebraic_int_real @ X2 )
% 0.29/0.48 => ( ( member_real @ X2 @ field_1537545994s_real )
% 0.29/0.48 => ( member_real @ X2 @ ring_1_Ints_real ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % rational_algebraic_int_is_int
% 0.29/0.48 thf(fact_243_algebraic__int__inverse,axiom,
% 0.29/0.48 ! [P: poly_real,X2: real] :
% 0.29/0.48 ( ( ( poly_real2 @ P @ X2 )
% 0.29/0.48 = zero_zero_real )
% 0.29/0.48 => ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
% 0.29/0.48 => ( ( ( coeff_real @ P @ zero_zero_nat )
% 0.29/0.48 = one_one_real )
% 0.29/0.48 => ( algebraic_int_real @ ( inverse_inverse_real @ X2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraic_int_inverse
% 0.29/0.48 thf(fact_244_algebraic__int__root,axiom,
% 0.29/0.48 ! [Y2: real,P: poly_real,X2: real] :
% 0.29/0.48 ( ( algebraic_int_real @ Y2 )
% 0.29/0.48 => ( ( ( poly_real2 @ P @ X2 )
% 0.29/0.48 = Y2 )
% 0.29/0.48 => ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
% 0.29/0.48 => ( ( ( coeff_real @ P @ ( degree_real @ P ) )
% 0.29/0.48 = one_one_real )
% 0.29/0.48 => ( ( ord_less_nat @ zero_zero_nat @ ( degree_real @ P ) )
% 0.29/0.48 => ( algebraic_int_real @ X2 ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % algebraic_int_root
% 0.29/0.48 thf(fact_245_gcd__nat_Oextremum__uniqueI,axiom,
% 0.29/0.48 ! [A3: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 0.29/0.48 => ( A3 = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.extremum_uniqueI
% 0.29/0.48 thf(fact_246_gcd__nat_Onot__eq__extremum,axiom,
% 0.29/0.48 ! [A3: nat] :
% 0.29/0.48 ( ( A3 != zero_zero_nat )
% 0.29/0.48 = ( ( dvd_dvd_nat @ A3 @ zero_zero_nat )
% 0.29/0.48 & ( A3 != zero_zero_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.not_eq_extremum
% 0.29/0.48 thf(fact_247_finite__Collect__less__nat,axiom,
% 0.29/0.48 ! [K: nat] :
% 0.29/0.48 ( finite_finite_nat
% 0.29/0.48 @ ( collect_nat
% 0.29/0.48 @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_Collect_less_nat
% 0.29/0.48 thf(fact_248_not__gr__zero,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.29/0.48 = ( N = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % not_gr_zero
% 0.29/0.48 thf(fact_249_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.48 ! [A3: nat] :
% 0.29/0.48 ( ( A3 != zero_zero_nat )
% 0.29/0.48 = ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot_nat_0.not_eq_extremum
% 0.29/0.48 thf(fact_250_less__nat__zero__code,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % less_nat_zero_code
% 0.29/0.48 thf(fact_251_neq0__conv,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( N != zero_zero_nat )
% 0.29/0.48 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % neq0_conv
% 0.29/0.48 thf(fact_252_lessI,axiom,
% 0.29/0.48 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % lessI
% 0.29/0.48 thf(fact_253_Suc__mono,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M @ N )
% 0.29/0.48 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Suc_mono
% 0.29/0.48 thf(fact_254_Suc__less__eq,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.48 = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % Suc_less_eq
% 0.29/0.48 thf(fact_255_zero__less__Suc,axiom,
% 0.29/0.48 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % zero_less_Suc
% 0.29/0.48 thf(fact_256_less__Suc0,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.48 = ( N = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_Suc0
% 0.29/0.48 thf(fact_257_less__one,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ N @ one_one_nat )
% 0.29/0.48 = ( N = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_one
% 0.29/0.48 thf(fact_258_zero__less__iff__neq__zero,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.48 = ( N != zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % zero_less_iff_neq_zero
% 0.29/0.48 thf(fact_259_gr__implies__not__zero,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M @ N )
% 0.29/0.48 => ( N != zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % gr_implies_not_zero
% 0.29/0.48 thf(fact_260_not__less__zero,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % not_less_zero
% 0.29/0.48 thf(fact_261_gr__zeroI,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( N != zero_zero_nat )
% 0.29/0.48 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % gr_zeroI
% 0.29/0.48 thf(fact_262_gr0I,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( N != zero_zero_nat )
% 0.29/0.48 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % gr0I
% 0.29/0.48 thf(fact_263_not__gr0,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.29/0.48 = ( N = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % not_gr0
% 0.29/0.48 thf(fact_264_not__less0,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % not_less0
% 0.29/0.48 thf(fact_265_less__zeroE,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % less_zeroE
% 0.29/0.48 thf(fact_266_gr__implies__not0,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M @ N )
% 0.29/0.48 => ( N != zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % gr_implies_not0
% 0.29/0.48 thf(fact_267_infinite__descent0,axiom,
% 0.29/0.48 ! [P2: nat > $o,N: nat] :
% 0.29/0.48 ( ( P2 @ zero_zero_nat )
% 0.29/0.48 => ( ! [N2: nat] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.48 => ( ~ ( P2 @ N2 )
% 0.29/0.48 => ? [M3: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M3 @ N2 )
% 0.29/0.48 & ~ ( P2 @ M3 ) ) ) )
% 0.29/0.48 => ( P2 @ N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % infinite_descent0
% 0.29/0.48 thf(fact_268_bot__nat__0_Oextremum__strict,axiom,
% 0.29/0.48 ! [A3: nat] :
% 0.29/0.48 ~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % bot_nat_0.extremum_strict
% 0.29/0.48 thf(fact_269_not__less__less__Suc__eq,axiom,
% 0.29/0.48 ! [N: nat,M: nat] :
% 0.29/0.48 ( ~ ( ord_less_nat @ N @ M )
% 0.29/0.48 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.29/0.48 = ( N = M ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % not_less_less_Suc_eq
% 0.29/0.48 thf(fact_270_strict__inc__induct,axiom,
% 0.29/0.48 ! [I2: nat,J: nat,P2: nat > $o] :
% 0.29/0.48 ( ( ord_less_nat @ I2 @ J )
% 0.29/0.48 => ( ! [I3: nat] :
% 0.29/0.48 ( ( J
% 0.29/0.48 = ( suc @ I3 ) )
% 0.29/0.48 => ( P2 @ I3 ) )
% 0.29/0.48 => ( ! [I3: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I3 @ J )
% 0.29/0.48 => ( ( P2 @ ( suc @ I3 ) )
% 0.29/0.48 => ( P2 @ I3 ) ) )
% 0.29/0.48 => ( P2 @ I2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % strict_inc_induct
% 0.29/0.48 thf(fact_271_less__Suc__induct,axiom,
% 0.29/0.48 ! [I2: nat,J: nat,P2: nat > nat > $o] :
% 0.29/0.48 ( ( ord_less_nat @ I2 @ J )
% 0.29/0.48 => ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
% 0.29/0.48 => ( ! [I3: nat,J2: nat,K2: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I3 @ J2 )
% 0.29/0.48 => ( ( ord_less_nat @ J2 @ K2 )
% 0.29/0.48 => ( ( P2 @ I3 @ J2 )
% 0.29/0.48 => ( ( P2 @ J2 @ K2 )
% 0.29/0.48 => ( P2 @ I3 @ K2 ) ) ) ) )
% 0.29/0.48 => ( P2 @ I2 @ J ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_Suc_induct
% 0.29/0.48 thf(fact_272_less__trans__Suc,axiom,
% 0.29/0.48 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I2 @ J )
% 0.29/0.48 => ( ( ord_less_nat @ J @ K )
% 0.29/0.48 => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_trans_Suc
% 0.29/0.48 thf(fact_273_Suc__less__SucD,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.48 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % Suc_less_SucD
% 0.29/0.48 thf(fact_274_less__antisym,axiom,
% 0.29/0.48 ! [N: nat,M: nat] :
% 0.29/0.48 ( ~ ( ord_less_nat @ N @ M )
% 0.29/0.48 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.29/0.48 => ( M = N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_antisym
% 0.29/0.48 thf(fact_275_Suc__less__eq2,axiom,
% 0.29/0.48 ! [N: nat,M: nat] :
% 0.29/0.48 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.29/0.48 = ( ? [M4: nat] :
% 0.29/0.48 ( ( M
% 0.29/0.48 = ( suc @ M4 ) )
% 0.29/0.48 & ( ord_less_nat @ N @ M4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Suc_less_eq2
% 0.29/0.48 thf(fact_276_All__less__Suc,axiom,
% 0.29/0.48 ! [N: nat,P2: nat > $o] :
% 0.29/0.48 ( ( ! [I4: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.29/0.48 => ( P2 @ I4 ) ) )
% 0.29/0.48 = ( ( P2 @ N )
% 0.29/0.48 & ! [I4: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I4 @ N )
% 0.29/0.48 => ( P2 @ I4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % All_less_Suc
% 0.29/0.48 thf(fact_277_not__less__eq,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.29/0.48 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % not_less_eq
% 0.29/0.48 thf(fact_278_less__Suc__eq,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.48 = ( ( ord_less_nat @ M @ N )
% 0.29/0.48 | ( M = N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_Suc_eq
% 0.29/0.48 thf(fact_279_Ex__less__Suc,axiom,
% 0.29/0.48 ! [N: nat,P2: nat > $o] :
% 0.29/0.48 ( ( ? [I4: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.29/0.48 & ( P2 @ I4 ) ) )
% 0.29/0.48 = ( ( P2 @ N )
% 0.29/0.48 | ? [I4: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I4 @ N )
% 0.29/0.48 & ( P2 @ I4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Ex_less_Suc
% 0.29/0.48 thf(fact_280_less__SucI,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M @ N )
% 0.29/0.48 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_SucI
% 0.29/0.48 thf(fact_281_less__SucE,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.48 => ( ~ ( ord_less_nat @ M @ N )
% 0.29/0.48 => ( M = N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_SucE
% 0.29/0.48 thf(fact_282_Suc__lessI,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M @ N )
% 0.29/0.48 => ( ( ( suc @ M )
% 0.29/0.48 != N )
% 0.29/0.48 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Suc_lessI
% 0.29/0.48 thf(fact_283_Suc__lessE,axiom,
% 0.29/0.48 ! [I2: nat,K: nat] :
% 0.29/0.48 ( ( ord_less_nat @ ( suc @ I2 ) @ K )
% 0.29/0.48 => ~ ! [J2: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I2 @ J2 )
% 0.29/0.48 => ( K
% 0.29/0.48 != ( suc @ J2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Suc_lessE
% 0.29/0.48 thf(fact_284_Suc__lessD,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.29/0.48 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % Suc_lessD
% 0.29/0.48 thf(fact_285_Nat_OlessE,axiom,
% 0.29/0.48 ! [I2: nat,K: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48 => ( ( K
% 0.29/0.48 != ( suc @ I2 ) )
% 0.29/0.48 => ~ ! [J2: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I2 @ J2 )
% 0.29/0.48 => ( K
% 0.29/0.48 != ( suc @ J2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Nat.lessE
% 0.29/0.48 thf(fact_286_Rats__inverse,axiom,
% 0.29/0.48 ! [A3: real] :
% 0.29/0.48 ( ( member_real @ A3 @ field_1537545994s_real )
% 0.29/0.48 => ( member_real @ ( inverse_inverse_real @ A3 ) @ field_1537545994s_real ) ) ).
% 0.29/0.48
% 0.29/0.48 % Rats_inverse
% 0.29/0.48 thf(fact_287_nat__neq__iff,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( M != N )
% 0.29/0.48 = ( ( ord_less_nat @ M @ N )
% 0.29/0.48 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat_neq_iff
% 0.29/0.48 thf(fact_288_less__not__refl,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.48
% 0.29/0.48 % less_not_refl
% 0.29/0.48 thf(fact_289_less__not__refl2,axiom,
% 0.29/0.48 ! [N: nat,M: nat] :
% 0.29/0.48 ( ( ord_less_nat @ N @ M )
% 0.29/0.48 => ( M != N ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_not_refl2
% 0.29/0.48 thf(fact_290_less__not__refl3,axiom,
% 0.29/0.48 ! [S: nat,T: nat] :
% 0.29/0.48 ( ( ord_less_nat @ S @ T )
% 0.29/0.48 => ( S != T ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_not_refl3
% 0.29/0.48 thf(fact_291_less__irrefl__nat,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.48
% 0.29/0.48 % less_irrefl_nat
% 0.29/0.48 thf(fact_292_nat__less__induct,axiom,
% 0.29/0.48 ! [P2: nat > $o,N: nat] :
% 0.29/0.48 ( ! [N2: nat] :
% 0.29/0.48 ( ! [M3: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M3 @ N2 )
% 0.29/0.48 => ( P2 @ M3 ) )
% 0.29/0.48 => ( P2 @ N2 ) )
% 0.29/0.48 => ( P2 @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat_less_induct
% 0.29/0.48 thf(fact_293_infinite__descent,axiom,
% 0.29/0.48 ! [P2: nat > $o,N: nat] :
% 0.29/0.48 ( ! [N2: nat] :
% 0.29/0.48 ( ~ ( P2 @ N2 )
% 0.29/0.48 => ? [M3: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M3 @ N2 )
% 0.29/0.48 & ~ ( P2 @ M3 ) ) )
% 0.29/0.48 => ( P2 @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % infinite_descent
% 0.29/0.48 thf(fact_294_linorder__neqE__nat,axiom,
% 0.29/0.48 ! [X2: nat,Y2: nat] :
% 0.29/0.48 ( ( X2 != Y2 )
% 0.29/0.48 => ( ~ ( ord_less_nat @ X2 @ Y2 )
% 0.29/0.48 => ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % linorder_neqE_nat
% 0.29/0.48 thf(fact_295_linorder__neqE__linordered__idom,axiom,
% 0.29/0.48 ! [X2: real,Y2: real] :
% 0.29/0.48 ( ( X2 != Y2 )
% 0.29/0.48 => ( ~ ( ord_less_real @ X2 @ Y2 )
% 0.29/0.48 => ( ord_less_real @ Y2 @ X2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % linorder_neqE_linordered_idom
% 0.29/0.48 thf(fact_296_lift__Suc__mono__less__iff,axiom,
% 0.29/0.48 ! [F: nat > nat,N: nat,M: nat] :
% 0.29/0.48 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.48 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.29/0.48 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % lift_Suc_mono_less_iff
% 0.29/0.48 thf(fact_297_lift__Suc__mono__less__iff,axiom,
% 0.29/0.48 ! [F: nat > real,N: nat,M: nat] :
% 0.29/0.48 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.48 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.29/0.48 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % lift_Suc_mono_less_iff
% 0.29/0.48 thf(fact_298_lift__Suc__mono__less,axiom,
% 0.29/0.48 ! [F: nat > nat,N: nat,N4: nat] :
% 0.29/0.48 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.48 => ( ( ord_less_nat @ N @ N4 )
% 0.29/0.48 => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % lift_Suc_mono_less
% 0.29/0.48 thf(fact_299_lift__Suc__mono__less,axiom,
% 0.29/0.48 ! [F: nat > real,N: nat,N4: nat] :
% 0.29/0.48 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.48 => ( ( ord_less_nat @ N @ N4 )
% 0.29/0.48 => ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % lift_Suc_mono_less
% 0.29/0.48 thf(fact_300_dvd__pos__nat,axiom,
% 0.29/0.48 ! [N: nat,M: nat] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.48 => ( ( dvd_dvd_nat @ M @ N )
% 0.29/0.48 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dvd_pos_nat
% 0.29/0.48 thf(fact_301_finite__divisors__nat,axiom,
% 0.29/0.48 ! [M: nat] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.48 => ( finite_finite_nat
% 0.29/0.48 @ ( collect_nat
% 0.29/0.48 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_divisors_nat
% 0.29/0.48 thf(fact_302_not__one__less__zero,axiom,
% 0.29/0.48 ~ ( ord_less_poly_real @ one_one_poly_real @ zero_zero_poly_real ) ).
% 0.29/0.48
% 0.29/0.48 % not_one_less_zero
% 0.29/0.48 thf(fact_303_not__one__less__zero,axiom,
% 0.29/0.48 ~ ( ord_le38482960y_real @ one_on501200385y_real @ zero_z1423781445y_real ) ).
% 0.29/0.48
% 0.29/0.48 % not_one_less_zero
% 0.29/0.48 thf(fact_304_not__one__less__zero,axiom,
% 0.29/0.48 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % not_one_less_zero
% 0.29/0.48 thf(fact_305_not__one__less__zero,axiom,
% 0.29/0.48 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.29/0.48
% 0.29/0.48 % not_one_less_zero
% 0.29/0.48 thf(fact_306_zero__less__one,axiom,
% 0.29/0.48 ord_less_poly_real @ zero_zero_poly_real @ one_one_poly_real ).
% 0.29/0.48
% 0.29/0.48 % zero_less_one
% 0.29/0.48 thf(fact_307_zero__less__one,axiom,
% 0.29/0.48 ord_le38482960y_real @ zero_z1423781445y_real @ one_on501200385y_real ).
% 0.29/0.48
% 0.29/0.48 % zero_less_one
% 0.29/0.48 thf(fact_308_zero__less__one,axiom,
% 0.29/0.48 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.29/0.48
% 0.29/0.48 % zero_less_one
% 0.29/0.48 thf(fact_309_zero__less__one,axiom,
% 0.29/0.48 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.29/0.48
% 0.29/0.48 % zero_less_one
% 0.29/0.48 thf(fact_310_less__Suc__eq__0__disj,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.48 = ( ( M = zero_zero_nat )
% 0.29/0.48 | ? [J3: nat] :
% 0.29/0.48 ( ( M
% 0.29/0.48 = ( suc @ J3 ) )
% 0.29/0.48 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_Suc_eq_0_disj
% 0.29/0.48 thf(fact_311_gr0__implies__Suc,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.48 => ? [M2: nat] :
% 0.29/0.48 ( N
% 0.29/0.48 = ( suc @ M2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gr0_implies_Suc
% 0.29/0.48 thf(fact_312_All__less__Suc2,axiom,
% 0.29/0.48 ! [N: nat,P2: nat > $o] :
% 0.29/0.48 ( ( ! [I4: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.29/0.48 => ( P2 @ I4 ) ) )
% 0.29/0.48 = ( ( P2 @ zero_zero_nat )
% 0.29/0.48 & ! [I4: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I4 @ N )
% 0.29/0.48 => ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % All_less_Suc2
% 0.29/0.48 thf(fact_313_gr0__conv__Suc,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.48 = ( ? [M5: nat] :
% 0.29/0.48 ( N
% 0.29/0.48 = ( suc @ M5 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gr0_conv_Suc
% 0.29/0.48 thf(fact_314_Ex__less__Suc2,axiom,
% 0.29/0.48 ! [N: nat,P2: nat > $o] :
% 0.29/0.48 ( ( ? [I4: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.29/0.48 & ( P2 @ I4 ) ) )
% 0.29/0.48 = ( ( P2 @ zero_zero_nat )
% 0.29/0.48 | ? [I4: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I4 @ N )
% 0.29/0.48 & ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Ex_less_Suc2
% 0.29/0.48 thf(fact_315_nat__dvd__not__less,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.48 => ( ( ord_less_nat @ M @ N )
% 0.29/0.48 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat_dvd_not_less
% 0.29/0.48 thf(fact_316_nat__induct__non__zero,axiom,
% 0.29/0.48 ! [N: nat,P2: nat > $o] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.48 => ( ( P2 @ one_one_nat )
% 0.29/0.48 => ( ! [N2: nat] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.48 => ( ( P2 @ N2 )
% 0.29/0.48 => ( P2 @ ( suc @ N2 ) ) ) )
% 0.29/0.48 => ( P2 @ N ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat_induct_non_zero
% 0.29/0.48 thf(fact_317_less__degree__imp,axiom,
% 0.29/0.48 ! [N: nat,P: poly_poly_real] :
% 0.29/0.48 ( ( ord_less_nat @ N @ ( degree_poly_real @ P ) )
% 0.29/0.48 => ? [I3: nat] :
% 0.29/0.48 ( ( ord_less_nat @ N @ I3 )
% 0.29/0.48 & ( ( coeff_poly_real @ P @ I3 )
% 0.29/0.48 != zero_zero_poly_real ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_degree_imp
% 0.29/0.48 thf(fact_318_less__degree__imp,axiom,
% 0.29/0.48 ! [N: nat,P: poly_nat] :
% 0.29/0.48 ( ( ord_less_nat @ N @ ( degree_nat @ P ) )
% 0.29/0.48 => ? [I3: nat] :
% 0.29/0.48 ( ( ord_less_nat @ N @ I3 )
% 0.29/0.48 & ( ( coeff_nat @ P @ I3 )
% 0.29/0.48 != zero_zero_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_degree_imp
% 0.29/0.48 thf(fact_319_less__degree__imp,axiom,
% 0.29/0.48 ! [N: nat,P: poly_poly_poly_real] :
% 0.29/0.48 ( ( ord_less_nat @ N @ ( degree360860553y_real @ P ) )
% 0.29/0.48 => ? [I3: nat] :
% 0.29/0.48 ( ( ord_less_nat @ N @ I3 )
% 0.29/0.48 & ( ( coeff_poly_poly_real @ P @ I3 )
% 0.29/0.48 != zero_z1423781445y_real ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_degree_imp
% 0.29/0.48 thf(fact_320_less__degree__imp,axiom,
% 0.29/0.48 ! [N: nat,P: poly_poly_nat] :
% 0.29/0.48 ( ( ord_less_nat @ N @ ( degree_poly_nat @ P ) )
% 0.29/0.48 => ? [I3: nat] :
% 0.29/0.48 ( ( ord_less_nat @ N @ I3 )
% 0.29/0.48 & ( ( coeff_poly_nat @ P @ I3 )
% 0.29/0.48 != zero_zero_poly_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_degree_imp
% 0.29/0.48 thf(fact_321_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat] :
% 0.29/0.48 ( ( A3 != B2 )
% 0.29/0.48 => ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 => ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 & ( A3 != B2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.not_eq_order_implies_strict
% 0.29/0.48 thf(fact_322_gcd__nat_Ostrict__implies__not__eq,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat] :
% 0.29/0.48 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 & ( A3 != B2 ) )
% 0.29/0.48 => ( A3 != B2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.strict_implies_not_eq
% 0.29/0.48 thf(fact_323_gcd__nat_Ostrict__implies__order,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat] :
% 0.29/0.48 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 & ( A3 != B2 ) )
% 0.29/0.48 => ( dvd_dvd_nat @ A3 @ B2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.strict_implies_order
% 0.29/0.48 thf(fact_324_gcd__nat_Ostrict__iff__order,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat] :
% 0.29/0.48 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 & ( A3 != B2 ) )
% 0.29/0.48 = ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 & ( A3 != B2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.strict_iff_order
% 0.29/0.48 thf(fact_325_gcd__nat_Oorder__iff__strict,axiom,
% 0.29/0.48 ( dvd_dvd_nat
% 0.29/0.48 = ( ^ [A: nat,B3: nat] :
% 0.29/0.48 ( ( ( dvd_dvd_nat @ A @ B3 )
% 0.29/0.48 & ( A != B3 ) )
% 0.29/0.48 | ( A = B3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.order_iff_strict
% 0.29/0.48 thf(fact_326_gcd__nat_Ostrict__trans2,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat,C: nat] :
% 0.29/0.48 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 & ( A3 != B2 ) )
% 0.29/0.48 => ( ( dvd_dvd_nat @ B2 @ C )
% 0.29/0.48 => ( ( dvd_dvd_nat @ A3 @ C )
% 0.29/0.48 & ( A3 != C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.strict_trans2
% 0.29/0.48 thf(fact_327_gcd__nat_Ostrict__trans1,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat,C: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 => ( ( ( dvd_dvd_nat @ B2 @ C )
% 0.29/0.48 & ( B2 != C ) )
% 0.29/0.48 => ( ( dvd_dvd_nat @ A3 @ C )
% 0.29/0.48 & ( A3 != C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.strict_trans1
% 0.29/0.48 thf(fact_328_gcd__nat_Ostrict__trans,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat,C: nat] :
% 0.29/0.48 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 & ( A3 != B2 ) )
% 0.29/0.48 => ( ( ( dvd_dvd_nat @ B2 @ C )
% 0.29/0.48 & ( B2 != C ) )
% 0.29/0.48 => ( ( dvd_dvd_nat @ A3 @ C )
% 0.29/0.48 & ( A3 != C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.strict_trans
% 0.29/0.48 thf(fact_329_gcd__nat_Oantisym,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 => ( ( dvd_dvd_nat @ B2 @ A3 )
% 0.29/0.48 => ( A3 = B2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.antisym
% 0.29/0.48 thf(fact_330_gcd__nat_Oirrefl,axiom,
% 0.29/0.48 ! [A3: nat] :
% 0.29/0.48 ~ ( ( dvd_dvd_nat @ A3 @ A3 )
% 0.29/0.48 & ( A3 != A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.irrefl
% 0.29/0.48 thf(fact_331_gcd__nat_Oeq__iff,axiom,
% 0.29/0.48 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.29/0.48 = ( ^ [A: nat,B3: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ A @ B3 )
% 0.29/0.48 & ( dvd_dvd_nat @ B3 @ A ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.eq_iff
% 0.29/0.48 thf(fact_332_gcd__nat_Otrans,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat,C: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 => ( ( dvd_dvd_nat @ B2 @ C )
% 0.29/0.48 => ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.trans
% 0.29/0.48 thf(fact_333_gcd__nat_Orefl,axiom,
% 0.29/0.48 ! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.refl
% 0.29/0.48 thf(fact_334_gcd__nat_Oasym,axiom,
% 0.29/0.48 ! [A3: nat,B2: nat] :
% 0.29/0.48 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.29/0.48 & ( A3 != B2 ) )
% 0.29/0.48 => ~ ( ( dvd_dvd_nat @ B2 @ A3 )
% 0.29/0.48 & ( B2 != A3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.asym
% 0.29/0.48 thf(fact_335_gcd__nat_Oextremum,axiom,
% 0.29/0.48 ! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.extremum
% 0.29/0.48 thf(fact_336_gcd__nat_Oextremum__strict,axiom,
% 0.29/0.48 ! [A3: nat] :
% 0.29/0.48 ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 0.29/0.48 & ( zero_zero_nat != A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.extremum_strict
% 0.29/0.48 thf(fact_337_gcd__nat_Oextremum__unique,axiom,
% 0.29/0.48 ! [A3: nat] :
% 0.29/0.48 ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 0.29/0.48 = ( A3 = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % gcd_nat.extremum_unique
% 0.29/0.48 thf(fact_338_poly__IVT__pos,axiom,
% 0.29/0.48 ! [A3: real,B2: real,P: poly_real] :
% 0.29/0.48 ( ( ord_less_real @ A3 @ B2 )
% 0.29/0.48 => ( ( ord_less_real @ ( poly_real2 @ P @ A3 ) @ zero_zero_real )
% 0.29/0.48 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B2 ) )
% 0.29/0.48 => ? [X3: real] :
% 0.29/0.48 ( ( ord_less_real @ A3 @ X3 )
% 0.29/0.48 & ( ord_less_real @ X3 @ B2 )
% 0.29/0.48 & ( ( poly_real2 @ P @ X3 )
% 0.29/0.48 = zero_zero_real ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_IVT_pos
% 0.29/0.48 thf(fact_339_poly__IVT__neg,axiom,
% 0.29/0.48 ! [A3: real,B2: real,P: poly_real] :
% 0.29/0.48 ( ( ord_less_real @ A3 @ B2 )
% 0.29/0.48 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A3 ) )
% 0.29/0.48 => ( ( ord_less_real @ ( poly_real2 @ P @ B2 ) @ zero_zero_real )
% 0.29/0.48 => ? [X3: real] :
% 0.29/0.48 ( ( ord_less_real @ A3 @ X3 )
% 0.29/0.48 & ( ord_less_real @ X3 @ B2 )
% 0.29/0.48 & ( ( poly_real2 @ P @ X3 )
% 0.29/0.48 = zero_zero_real ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_IVT_neg
% 0.29/0.48 thf(fact_340_finite__M__bounded__by__nat,axiom,
% 0.29/0.48 ! [P2: nat > $o,I2: nat] :
% 0.29/0.48 ( finite_finite_nat
% 0.29/0.48 @ ( collect_nat
% 0.29/0.48 @ ^ [K3: nat] :
% 0.29/0.48 ( ( P2 @ K3 )
% 0.29/0.48 & ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_M_bounded_by_nat
% 0.29/0.48 thf(fact_341_Rats__dense__in__real,axiom,
% 0.29/0.48 ! [X2: real,Y2: real] :
% 0.29/0.48 ( ( ord_less_real @ X2 @ Y2 )
% 0.29/0.48 => ? [X3: real] :
% 0.29/0.48 ( ( member_real @ X3 @ field_1537545994s_real )
% 0.29/0.48 & ( ord_less_real @ X2 @ X3 )
% 0.29/0.48 & ( ord_less_real @ X3 @ Y2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Rats_dense_in_real
% 0.29/0.48 thf(fact_342_Rats__no__bot__less,axiom,
% 0.29/0.48 ! [X2: real] :
% 0.29/0.48 ? [X3: real] :
% 0.29/0.48 ( ( member_real @ X3 @ field_1537545994s_real )
% 0.29/0.48 & ( ord_less_real @ X3 @ X2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % Rats_no_bot_less
% 0.29/0.48 thf(fact_343_poly__pinfty__gt__lc,axiom,
% 0.29/0.48 ! [P: poly_real] :
% 0.29/0.48 ( ( ord_less_real @ zero_zero_real @ ( coeff_real @ P @ ( degree_real @ P ) ) )
% 0.29/0.48 => ? [N2: real] :
% 0.29/0.48 ! [X4: real] :
% 0.29/0.48 ( ( ord_less_eq_real @ N2 @ X4 )
% 0.29/0.48 => ( ord_less_eq_real @ ( coeff_real @ P @ ( degree_real @ P ) ) @ ( poly_real2 @ P @ X4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_pinfty_gt_lc
% 0.29/0.48
% 0.29/0.48 % Conjectures (1)
% 0.29/0.48 thf(conj_0,conjecture,
% 0.29/0.48 ( finite_finite_real
% 0.29/0.48 @ ( collect_real
% 0.29/0.48 @ ^ [X: real] :
% 0.29/0.48 ( ( poly_real2 @ ( pderiv_real @ p ) @ X )
% 0.29/0.48 = zero_zero_real ) ) ) ).
% 0.29/0.48
% 0.29/0.48 %------------------------------------------------------------------------------
% 0.29/0.48 ------------------------
% 2.52/1.51 ---- Embedded in THF ---
% 2.52/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.52/1.51 %%% Generated on Tue Feb 24 08:26:29 EST 2026
% 2.52/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 2.52/1.51 %%% Logic specification used:
% 2.52/1.51 %%% thf(spec, logic, $$dhol).
% 2.52/1.51
% 2.52/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2DTF11513.p
% See solution above
% 2.52/1.51 ------------------------
% 2.52/1.53 ---- Cleaned THF ---
% 2.52/1.53 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 2.52/1.53 poly_poly_poly_real: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 2.52/1.53 set_poly_poly_real: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 poly_poly_real: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 2.52/1.53 poly_poly_nat: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 set_poly_real: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 poly_real: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 2.52/1.53 poly_nat: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 2.52/1.53 set_real: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.52/1.53 set_nat: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Real__Oreal,type,
% 2.52/1.53 real: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(ty_n_t__Nat__Onat,type,
% 2.52/1.53 nat: $tType ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 2.52/1.53 inverse_inverse_real: real > real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 2.52/1.53 finite_finite_nat: set_nat > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Finite__Set_Ofinite_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 finite1328464339y_real: set_poly_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Finite__Set_Ofinite_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 finite1810960971y_real: set_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
% 2.52/1.53 finite_finite_real: set_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 2.52/1.53 one_one_nat: nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 2.52/1.53 one_one_poly_nat: poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 one_on501200385y_real: poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 one_one_poly_real: poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 2.52/1.53 one_one_real: real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.52/1.53 zero_zero_nat: nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 2.52/1.53 zero_zero_poly_nat: poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
% 2.52/1.53 zero_z1059985641ly_nat: poly_poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
% 2.52/1.53 zero_z935034829y_real: poly_poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 zero_z1423781445y_real: poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 zero_zero_poly_real: poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 2.52/1.53 zero_zero_real: real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Int_Oring__1__class_OInts_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 ring_1897377867y_real: set_poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Int_Oring__1__class_OInts_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 ring_1690226883y_real: set_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 2.52/1.53 ring_1_Ints_real: set_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Nat_OSuc,type,
% 2.52/1.53 suc: nat > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.52/1.53 ord_less_nat: nat > nat > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 ord_less_poly_real: poly_real > poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 2.52/1.53 ord_less_real: real > real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 2.52/1.53 ord_less_eq_real: real > real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oalgebraic_001t__Real__Oreal,type,
% 2.52/1.53 algebraic_real: real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oalgebraic__int_001t__Real__Oreal,type,
% 2.52/1.53 algebraic_int_real: real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Ocr__poly_001t__Real__Oreal,type,
% 2.52/1.53 cr_poly_real: ( nat > real ) > poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Odegree_001t__Nat__Onat,type,
% 2.52/1.53 degree_nat: poly_nat > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 2.52/1.53 degree_poly_nat: poly_poly_nat > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 degree360860553y_real: poly_poly_poly_real > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 degree_poly_real: poly_poly_real > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Odegree_001t__Real__Oreal,type,
% 2.52/1.53 degree_real: poly_real > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 divide924636027y_real: poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > nat > nat > poly_poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Odivide__poly__main_001t__Real__Oreal,type,
% 2.52/1.53 divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat,type,
% 2.52/1.53 is_zero_nat: poly_nat > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 is_zero_poly_real: poly_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Ois__zero_001t__Real__Oreal,type,
% 2.52/1.53 is_zero_real: poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 order_poly_real: poly_real > poly_poly_real > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oorder_001t__Real__Oreal,type,
% 2.52/1.53 order_real: real > poly_real > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opcr__poly_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.52/1.53 pcr_poly_nat_nat: ( nat > nat > $o ) > ( nat > nat ) > poly_nat > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opcr__poly_001t__Polynomial__Opoly_It__Nat__Onat_J_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 2.52/1.53 pcr_po273983709ly_nat: ( poly_nat > poly_nat > $o ) > ( nat > poly_nat ) > poly_poly_nat > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opcr__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 pcr_po1200519205y_real: ( poly_poly_real > poly_poly_real > $o ) > ( nat > poly_poly_real ) > poly_poly_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opcr__poly_001t__Polynomial__Opoly_It__Real__Oreal_J_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 pcr_po1314690837y_real: ( poly_real > poly_real > $o ) > ( nat > poly_real ) > poly_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opcr__poly_001t__Real__Oreal_001t__Real__Oreal,type,
% 2.52/1.53 pcr_poly_real_real: ( real > real > $o ) > ( nat > real ) > poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opderiv_001t__Nat__Onat,type,
% 2.52/1.53 pderiv_nat: poly_nat > poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opderiv_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 pderiv_poly_real: poly_poly_real > poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opderiv_001t__Real__Oreal,type,
% 2.52/1.53 pderiv_real: poly_real > poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
% 2.52/1.53 poly_nat2: poly_nat > nat > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 2.52/1.53 poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 poly_poly_real2: poly_poly_real > poly_real > poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
% 2.52/1.53 poly_real2: poly_real > real > real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat,type,
% 2.52/1.53 coeff_nat: poly_nat > nat > nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 2.52/1.53 coeff_poly_nat: poly_poly_nat > nat > poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 coeff_poly_poly_real: poly_poly_poly_real > nat > poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 coeff_poly_real: poly_poly_real > nat > poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Real__Oreal,type,
% 2.52/1.53 coeff_real: poly_real > nat > real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat,type,
% 2.52/1.53 poly_cutoff_nat: nat > poly_nat > poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly__cutoff_001t__Real__Oreal,type,
% 2.52/1.53 poly_cutoff_real: nat > poly_real > poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat,type,
% 2.52/1.53 poly_shift_nat: nat > poly_nat > poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Opoly__shift_001t__Real__Oreal,type,
% 2.52/1.53 poly_shift_real: nat > poly_real > poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
% 2.52/1.53 reflect_poly_nat: poly_nat > poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 2.52/1.53 reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 reflec1522834046y_real: poly_poly_real > poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal,type,
% 2.52/1.53 reflect_poly_real: poly_real > poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Orsquarefree_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 rsquar1555552848y_real: poly_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Orsquarefree_001t__Real__Oreal,type,
% 2.52/1.53 rsquarefree_real: poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Osynthetic__div_001t__Nat__Onat,type,
% 2.52/1.53 synthetic_div_nat: poly_nat > nat > poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Osynthetic__div_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 synthe1498897281y_real: poly_poly_real > poly_real > poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Polynomial_Osynthetic__div_001t__Real__Oreal,type,
% 2.52/1.53 synthetic_div_real: poly_real > real > poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 2.52/1.53 field_1537545994s_real: set_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 2.52/1.53 dvd_dvd_nat: nat > nat > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 2.52/1.53 dvd_dvd_poly_nat: poly_nat > poly_nat > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 dvd_dv1946063458y_real: poly_poly_real > poly_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 dvd_dvd_poly_real: poly_real > poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 2.52/1.53 dvd_dvd_real: real > real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 2.52/1.53 collect_nat: ( nat > $o ) > set_nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 collec927113489y_real: ( poly_poly_real > $o ) > set_poly_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 collect_poly_real: ( poly_real > $o ) > set_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 2.52/1.53 collect_real: ( real > $o ) > set_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_member_001t__Nat__Onat,type,
% 2.52/1.53 member_nat: nat > set_nat > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_member_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 2.52/1.53 member1159720147y_real: poly_poly_real > set_poly_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_member_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 2.52/1.53 member_poly_real: poly_real > set_poly_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_c_member_001t__Real__Oreal,type,
% 2.52/1.53 member_real: real > set_real > $o ).
% 2.52/1.53
% 2.52/1.53 thf(sy_v_n____,type,
% 2.52/1.53 n: nat ).
% 2.52/1.53
% 2.52/1.53 thf(sy_v_p____,type,
% 2.52/1.53 p: poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(sy_v_x,type,
% 2.52/1.53 x: real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_0_poly__roots__finite,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( P != zero_z935034829y_real )
% 2.52/1.53 => ( finite1328464339y_real
% 2.52/1.53 @ ( collec927113489y_real
% 2.52/1.53 @ ^ [X: poly_poly_real] :
% 2.52/1.53 ( ( poly_poly_poly_real2 @ P @ X )
% 2.52/1.53 = zero_z1423781445y_real ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_1_poly__roots__finite,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( P != zero_z1423781445y_real )
% 2.52/1.53 => ( finite1810960971y_real
% 2.52/1.53 @ ( collect_poly_real
% 2.52/1.53 @ ^ [X: poly_real] :
% 2.52/1.53 ( ( poly_poly_real2 @ P @ X )
% 2.52/1.53 = zero_zero_poly_real ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_2_poly__roots__finite,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( P != zero_zero_poly_real )
% 2.52/1.53 => ( finite_finite_real
% 2.52/1.53 @ ( collect_real
% 2.52/1.53 @ ^ [X: real] :
% 2.52/1.53 ( ( poly_real2 @ P @ X )
% 2.52/1.53 = zero_zero_real ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_3_p_I2_J,axiom,
% 2.52/1.53 p != zero_zero_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_4__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062,axiom,
% 2.52/1.53 ( ( pderiv_real @ p )
% 2.52/1.53 != zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_5_p_I3_J,axiom,
% 2.52/1.53 ( ( poly_real2 @ p @ x )
% 2.52/1.53 = zero_zero_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_6_poly__0,axiom,
% 2.52/1.53 ! [X2: poly_poly_real] :
% 2.52/1.53 ( ( poly_poly_poly_real2 @ zero_z935034829y_real @ X2 )
% 2.52/1.53 = zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_7_poly__0,axiom,
% 2.52/1.53 ! [X2: poly_nat] :
% 2.52/1.53 ( ( poly_poly_nat2 @ zero_z1059985641ly_nat @ X2 )
% 2.52/1.53 = zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_8_poly__0,axiom,
% 2.52/1.53 ! [X2: poly_real] :
% 2.52/1.53 ( ( poly_poly_real2 @ zero_z1423781445y_real @ X2 )
% 2.52/1.53 = zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_9_poly__0,axiom,
% 2.52/1.53 ! [X2: nat] :
% 2.52/1.53 ( ( poly_nat2 @ zero_zero_poly_nat @ X2 )
% 2.52/1.53 = zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_10_poly__0,axiom,
% 2.52/1.53 ! [X2: real] :
% 2.52/1.53 ( ( poly_real2 @ zero_zero_poly_real @ X2 )
% 2.52/1.53 = zero_zero_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_11_finite__Collect__conjI,axiom,
% 2.52/1.53 ! [P2: poly_real > $o,Q: poly_real > $o] :
% 2.52/1.53 ( ( ( finite1810960971y_real @ ( collect_poly_real @ P2 ) )
% 2.52/1.53 | ( finite1810960971y_real @ ( collect_poly_real @ Q ) ) )
% 2.52/1.53 => ( finite1810960971y_real
% 2.52/1.53 @ ( collect_poly_real
% 2.52/1.53 @ ^ [X: poly_real] :
% 2.52/1.53 ( ( P2 @ X )
% 2.52/1.53 & ( Q @ X ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_12_finite__Collect__conjI,axiom,
% 2.52/1.53 ! [P2: real > $o,Q: real > $o] :
% 2.52/1.53 ( ( ( finite_finite_real @ ( collect_real @ P2 ) )
% 2.52/1.53 | ( finite_finite_real @ ( collect_real @ Q ) ) )
% 2.52/1.53 => ( finite_finite_real
% 2.52/1.53 @ ( collect_real
% 2.52/1.53 @ ^ [X: real] :
% 2.52/1.53 ( ( P2 @ X )
% 2.52/1.53 & ( Q @ X ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_13_finite__Collect__conjI,axiom,
% 2.52/1.53 ! [P2: nat > $o,Q: nat > $o] :
% 2.52/1.53 ( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 2.52/1.53 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 2.52/1.53 => ( finite_finite_nat
% 2.52/1.53 @ ( collect_nat
% 2.52/1.53 @ ^ [X: nat] :
% 2.52/1.53 ( ( P2 @ X )
% 2.52/1.53 & ( Q @ X ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_14_finite__Collect__disjI,axiom,
% 2.52/1.53 ! [P2: poly_real > $o,Q: poly_real > $o] :
% 2.52/1.53 ( ( finite1810960971y_real
% 2.52/1.53 @ ( collect_poly_real
% 2.52/1.53 @ ^ [X: poly_real] :
% 2.52/1.53 ( ( P2 @ X )
% 2.52/1.53 | ( Q @ X ) ) ) )
% 2.52/1.53 = ( ( finite1810960971y_real @ ( collect_poly_real @ P2 ) )
% 2.52/1.53 & ( finite1810960971y_real @ ( collect_poly_real @ Q ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_15_finite__Collect__disjI,axiom,
% 2.52/1.53 ! [P2: real > $o,Q: real > $o] :
% 2.52/1.53 ( ( finite_finite_real
% 2.52/1.53 @ ( collect_real
% 2.52/1.53 @ ^ [X: real] :
% 2.52/1.53 ( ( P2 @ X )
% 2.52/1.53 | ( Q @ X ) ) ) )
% 2.52/1.53 = ( ( finite_finite_real @ ( collect_real @ P2 ) )
% 2.52/1.53 & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_16_finite__Collect__disjI,axiom,
% 2.52/1.53 ! [P2: nat > $o,Q: nat > $o] :
% 2.52/1.53 ( ( finite_finite_nat
% 2.52/1.53 @ ( collect_nat
% 2.52/1.53 @ ^ [X: nat] :
% 2.52/1.53 ( ( P2 @ X )
% 2.52/1.53 | ( Q @ X ) ) ) )
% 2.52/1.53 = ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 2.52/1.53 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_17_poly__all__0__iff__0,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( ! [X: poly_poly_real] :
% 2.52/1.53 ( ( poly_poly_poly_real2 @ P @ X )
% 2.52/1.53 = zero_z1423781445y_real ) )
% 2.52/1.53 = ( P = zero_z935034829y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_18_poly__all__0__iff__0,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( ! [X: real] :
% 2.52/1.53 ( ( poly_real2 @ P @ X )
% 2.52/1.53 = zero_zero_real ) )
% 2.52/1.53 = ( P = zero_zero_poly_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_19_poly__all__0__iff__0,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( ! [X: poly_real] :
% 2.52/1.53 ( ( poly_poly_real2 @ P @ X )
% 2.52/1.53 = zero_zero_poly_real ) )
% 2.52/1.53 = ( P = zero_z1423781445y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_20_rsquarefree__roots,axiom,
% 2.52/1.53 ( rsquarefree_real
% 2.52/1.53 = ( ^ [P3: poly_real] :
% 2.52/1.53 ! [A: real] :
% 2.52/1.53 ~ ( ( ( poly_real2 @ P3 @ A )
% 2.52/1.53 = zero_zero_real )
% 2.52/1.53 & ( ( poly_real2 @ ( pderiv_real @ P3 ) @ A )
% 2.52/1.53 = zero_zero_real ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_21_not__finite__existsD,axiom,
% 2.52/1.53 ! [P2: real > $o] :
% 2.52/1.53 ( ~ ( finite_finite_real @ ( collect_real @ P2 ) )
% 2.52/1.53 => ? [X_1: real] : ( P2 @ X_1 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_22_not__finite__existsD,axiom,
% 2.52/1.53 ! [P2: nat > $o] :
% 2.52/1.53 ( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
% 2.52/1.53 => ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_23_not__finite__existsD,axiom,
% 2.52/1.53 ! [P2: poly_real > $o] :
% 2.52/1.53 ( ~ ( finite1810960971y_real @ ( collect_poly_real @ P2 ) )
% 2.52/1.53 => ? [X_1: poly_real] : ( P2 @ X_1 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_24_pigeonhole__infinite__rel,axiom,
% 2.52/1.53 ! [A2: set_real,B: set_real,R: real > real > $o] :
% 2.52/1.53 ( ~ ( finite_finite_real @ A2 )
% 2.52/1.53 => ( ( finite_finite_real @ B )
% 2.52/1.53 => ( ! [X3: real] :
% 2.52/1.53 ( ( member_real @ X3 @ A2 )
% 2.52/1.53 => ? [Xa: real] :
% 2.52/1.53 ( ( member_real @ Xa @ B )
% 2.52/1.53 & ( R @ X3 @ Xa ) ) )
% 2.52/1.53 => ? [X3: real] :
% 2.52/1.53 ( ( member_real @ X3 @ B )
% 2.52/1.53 & ~ ( finite_finite_real
% 2.52/1.53 @ ( collect_real
% 2.52/1.53 @ ^ [A: real] :
% 2.52/1.53 ( ( member_real @ A @ A2 )
% 2.52/1.53 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_25_pigeonhole__infinite__rel,axiom,
% 2.52/1.53 ! [A2: set_real,B: set_nat,R: real > nat > $o] :
% 2.52/1.53 ( ~ ( finite_finite_real @ A2 )
% 2.52/1.53 => ( ( finite_finite_nat @ B )
% 2.52/1.53 => ( ! [X3: real] :
% 2.52/1.53 ( ( member_real @ X3 @ A2 )
% 2.52/1.53 => ? [Xa: nat] :
% 2.52/1.53 ( ( member_nat @ Xa @ B )
% 2.52/1.53 & ( R @ X3 @ Xa ) ) )
% 2.52/1.53 => ? [X3: nat] :
% 2.52/1.53 ( ( member_nat @ X3 @ B )
% 2.52/1.53 & ~ ( finite_finite_real
% 2.52/1.53 @ ( collect_real
% 2.52/1.53 @ ^ [A: real] :
% 2.52/1.53 ( ( member_real @ A @ A2 )
% 2.52/1.53 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_26_pigeonhole__infinite__rel,axiom,
% 2.52/1.53 ! [A2: set_real,B: set_poly_real,R: real > poly_real > $o] :
% 2.52/1.53 ( ~ ( finite_finite_real @ A2 )
% 2.52/1.53 => ( ( finite1810960971y_real @ B )
% 2.52/1.53 => ( ! [X3: real] :
% 2.52/1.53 ( ( member_real @ X3 @ A2 )
% 2.52/1.53 => ? [Xa: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ Xa @ B )
% 2.52/1.53 & ( R @ X3 @ Xa ) ) )
% 2.52/1.53 => ? [X3: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ X3 @ B )
% 2.52/1.53 & ~ ( finite_finite_real
% 2.52/1.53 @ ( collect_real
% 2.52/1.53 @ ^ [A: real] :
% 2.52/1.53 ( ( member_real @ A @ A2 )
% 2.52/1.53 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_27_pigeonhole__infinite__rel,axiom,
% 2.52/1.53 ! [A2: set_nat,B: set_real,R: nat > real > $o] :
% 2.52/1.53 ( ~ ( finite_finite_nat @ A2 )
% 2.52/1.53 => ( ( finite_finite_real @ B )
% 2.52/1.53 => ( ! [X3: nat] :
% 2.52/1.53 ( ( member_nat @ X3 @ A2 )
% 2.52/1.53 => ? [Xa: real] :
% 2.52/1.53 ( ( member_real @ Xa @ B )
% 2.52/1.53 & ( R @ X3 @ Xa ) ) )
% 2.52/1.53 => ? [X3: real] :
% 2.52/1.53 ( ( member_real @ X3 @ B )
% 2.52/1.53 & ~ ( finite_finite_nat
% 2.52/1.53 @ ( collect_nat
% 2.52/1.53 @ ^ [A: nat] :
% 2.52/1.53 ( ( member_nat @ A @ A2 )
% 2.52/1.53 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_28_pigeonhole__infinite__rel,axiom,
% 2.52/1.53 ! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
% 2.52/1.53 ( ~ ( finite_finite_nat @ A2 )
% 2.52/1.53 => ( ( finite_finite_nat @ B )
% 2.52/1.53 => ( ! [X3: nat] :
% 2.52/1.53 ( ( member_nat @ X3 @ A2 )
% 2.52/1.53 => ? [Xa: nat] :
% 2.52/1.53 ( ( member_nat @ Xa @ B )
% 2.52/1.53 & ( R @ X3 @ Xa ) ) )
% 2.52/1.53 => ? [X3: nat] :
% 2.52/1.53 ( ( member_nat @ X3 @ B )
% 2.52/1.53 & ~ ( finite_finite_nat
% 2.52/1.53 @ ( collect_nat
% 2.52/1.53 @ ^ [A: nat] :
% 2.52/1.53 ( ( member_nat @ A @ A2 )
% 2.52/1.53 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_29_pigeonhole__infinite__rel,axiom,
% 2.52/1.53 ! [A2: set_nat,B: set_poly_real,R: nat > poly_real > $o] :
% 2.52/1.53 ( ~ ( finite_finite_nat @ A2 )
% 2.52/1.53 => ( ( finite1810960971y_real @ B )
% 2.52/1.53 => ( ! [X3: nat] :
% 2.52/1.53 ( ( member_nat @ X3 @ A2 )
% 2.52/1.53 => ? [Xa: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ Xa @ B )
% 2.52/1.53 & ( R @ X3 @ Xa ) ) )
% 2.52/1.53 => ? [X3: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ X3 @ B )
% 2.52/1.53 & ~ ( finite_finite_nat
% 2.52/1.53 @ ( collect_nat
% 2.52/1.53 @ ^ [A: nat] :
% 2.52/1.53 ( ( member_nat @ A @ A2 )
% 2.52/1.53 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_30_pigeonhole__infinite__rel,axiom,
% 2.52/1.53 ! [A2: set_poly_real,B: set_real,R: poly_real > real > $o] :
% 2.52/1.53 ( ~ ( finite1810960971y_real @ A2 )
% 2.52/1.53 => ( ( finite_finite_real @ B )
% 2.52/1.53 => ( ! [X3: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ X3 @ A2 )
% 2.52/1.53 => ? [Xa: real] :
% 2.52/1.53 ( ( member_real @ Xa @ B )
% 2.52/1.53 & ( R @ X3 @ Xa ) ) )
% 2.52/1.53 => ? [X3: real] :
% 2.52/1.53 ( ( member_real @ X3 @ B )
% 2.52/1.53 & ~ ( finite1810960971y_real
% 2.52/1.53 @ ( collect_poly_real
% 2.52/1.53 @ ^ [A: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ A @ A2 )
% 2.52/1.53 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_31_pigeonhole__infinite__rel,axiom,
% 2.52/1.53 ! [A2: set_poly_real,B: set_nat,R: poly_real > nat > $o] :
% 2.52/1.53 ( ~ ( finite1810960971y_real @ A2 )
% 2.52/1.53 => ( ( finite_finite_nat @ B )
% 2.52/1.53 => ( ! [X3: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ X3 @ A2 )
% 2.52/1.53 => ? [Xa: nat] :
% 2.52/1.53 ( ( member_nat @ Xa @ B )
% 2.52/1.53 & ( R @ X3 @ Xa ) ) )
% 2.52/1.53 => ? [X3: nat] :
% 2.52/1.53 ( ( member_nat @ X3 @ B )
% 2.52/1.53 & ~ ( finite1810960971y_real
% 2.52/1.53 @ ( collect_poly_real
% 2.52/1.53 @ ^ [A: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ A @ A2 )
% 2.52/1.53 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_32_pigeonhole__infinite__rel,axiom,
% 2.52/1.53 ! [A2: set_poly_real,B: set_poly_real,R: poly_real > poly_real > $o] :
% 2.52/1.53 ( ~ ( finite1810960971y_real @ A2 )
% 2.52/1.53 => ( ( finite1810960971y_real @ B )
% 2.52/1.53 => ( ! [X3: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ X3 @ A2 )
% 2.52/1.53 => ? [Xa: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ Xa @ B )
% 2.52/1.53 & ( R @ X3 @ Xa ) ) )
% 2.52/1.53 => ? [X3: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ X3 @ B )
% 2.52/1.53 & ~ ( finite1810960971y_real
% 2.52/1.53 @ ( collect_poly_real
% 2.52/1.53 @ ^ [A: poly_real] :
% 2.52/1.53 ( ( member_poly_real @ A @ A2 )
% 2.52/1.53 & ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_33_poly__eq__poly__eq__iff,axiom,
% 2.52/1.53 ! [P: poly_real,Q2: poly_real] :
% 2.52/1.53 ( ( ( poly_real2 @ P )
% 2.52/1.53 = ( poly_real2 @ Q2 ) )
% 2.52/1.53 = ( P = Q2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_34_poly__eq__poly__eq__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_real,Q2: poly_poly_real] :
% 2.52/1.53 ( ( ( poly_poly_real2 @ P )
% 2.52/1.53 = ( poly_poly_real2 @ Q2 ) )
% 2.52/1.53 = ( P = Q2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_35_n__def,axiom,
% 2.52/1.53 ( n
% 2.52/1.53 = ( degree_real @ p ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_36_assms_I2_J,axiom,
% 2.52/1.53 algebraic_real @ x ).
% 2.52/1.53
% 2.52/1.53 thf(fact_37_irrationsl,axiom,
% 2.52/1.53 ~ ( member_real @ x @ field_1537545994s_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_38_degree__0,axiom,
% 2.52/1.53 ( ( degree_real @ zero_zero_poly_real )
% 2.52/1.53 = zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_39_degree__0,axiom,
% 2.52/1.53 ( ( degree_poly_real @ zero_z1423781445y_real )
% 2.52/1.53 = zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_40_degree__0,axiom,
% 2.52/1.53 ( ( degree_nat @ zero_zero_poly_nat )
% 2.52/1.53 = zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_41_pderiv__0,axiom,
% 2.52/1.53 ( ( pderiv_real @ zero_zero_poly_real )
% 2.52/1.53 = zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_42_pderiv__0,axiom,
% 2.52/1.53 ( ( pderiv_poly_real @ zero_z1423781445y_real )
% 2.52/1.53 = zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_43_pderiv__0,axiom,
% 2.52/1.53 ( ( pderiv_nat @ zero_zero_poly_nat )
% 2.52/1.53 = zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_44_pderiv__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( ( pderiv_real @ P )
% 2.52/1.53 = zero_zero_poly_real )
% 2.52/1.53 = ( ( degree_real @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_45_pderiv__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( ( pderiv_poly_real @ P )
% 2.52/1.53 = zero_z1423781445y_real )
% 2.52/1.53 = ( ( degree_poly_real @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_46_pderiv__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_nat] :
% 2.52/1.53 ( ( ( pderiv_nat @ P )
% 2.52/1.53 = zero_zero_poly_nat )
% 2.52/1.53 = ( ( degree_nat @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_47_zero__reorient,axiom,
% 2.52/1.53 ! [X2: real] :
% 2.52/1.53 ( ( zero_zero_real = X2 )
% 2.52/1.53 = ( X2 = zero_zero_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_48_zero__reorient,axiom,
% 2.52/1.53 ! [X2: poly_real] :
% 2.52/1.53 ( ( zero_zero_poly_real = X2 )
% 2.52/1.53 = ( X2 = zero_zero_poly_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_49_zero__reorient,axiom,
% 2.52/1.53 ! [X2: nat] :
% 2.52/1.53 ( ( zero_zero_nat = X2 )
% 2.52/1.53 = ( X2 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_50_zero__reorient,axiom,
% 2.52/1.53 ! [X2: poly_poly_real] :
% 2.52/1.53 ( ( zero_z1423781445y_real = X2 )
% 2.52/1.53 = ( X2 = zero_z1423781445y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_51_zero__reorient,axiom,
% 2.52/1.53 ! [X2: poly_nat] :
% 2.52/1.53 ( ( zero_zero_poly_nat = X2 )
% 2.52/1.53 = ( X2 = zero_zero_poly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_52_is__zero__null,axiom,
% 2.52/1.53 ( is_zero_real
% 2.52/1.53 = ( ^ [P3: poly_real] : ( P3 = zero_zero_poly_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_53_is__zero__null,axiom,
% 2.52/1.53 ( is_zero_poly_real
% 2.52/1.53 = ( ^ [P3: poly_poly_real] : ( P3 = zero_z1423781445y_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_54_is__zero__null,axiom,
% 2.52/1.53 ( is_zero_nat
% 2.52/1.53 = ( ^ [P3: poly_nat] : ( P3 = zero_zero_poly_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_55_poly__cutoff__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( poly_cutoff_real @ N @ zero_zero_poly_real )
% 2.52/1.53 = zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_56_poly__cutoff__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( poly_c1404107022y_real @ N @ zero_z1423781445y_real )
% 2.52/1.53 = zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_57_poly__cutoff__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
% 2.52/1.53 = zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_58_reflect__poly__at__0__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
% 2.52/1.53 = zero_zero_real )
% 2.52/1.53 = ( P = zero_zero_poly_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_59_reflect__poly__at__0__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
% 2.52/1.53 = zero_zero_poly_real )
% 2.52/1.53 = ( P = zero_z1423781445y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_60_reflect__poly__at__0__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_nat] :
% 2.52/1.53 ( ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 2.52/1.53 = zero_zero_nat )
% 2.52/1.53 = ( P = zero_zero_poly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_61_reflect__poly__at__0__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
% 2.52/1.53 = zero_z1423781445y_real )
% 2.52/1.53 = ( P = zero_z935034829y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_62_reflect__poly__at__0__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_nat] :
% 2.52/1.53 ( ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
% 2.52/1.53 = zero_zero_poly_nat )
% 2.52/1.53 = ( P = zero_z1059985641ly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_63__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062,axiom,
% 2.52/1.53 ~ ! [P4: poly_real] :
% 2.52/1.53 ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
% 2.52/1.53 => ( ( P4 != zero_zero_poly_real )
% 2.52/1.53 => ( ( poly_real2 @ P4 @ x )
% 2.52/1.53 != zero_zero_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_64_synthetic__div__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_real,C: real] :
% 2.52/1.53 ( ( ( synthetic_div_real @ P @ C )
% 2.52/1.53 = zero_zero_poly_real )
% 2.52/1.53 = ( ( degree_real @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_65_synthetic__div__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_real,C: poly_real] :
% 2.52/1.53 ( ( ( synthe1498897281y_real @ P @ C )
% 2.52/1.53 = zero_z1423781445y_real )
% 2.52/1.53 = ( ( degree_poly_real @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_66_synthetic__div__eq__0__iff,axiom,
% 2.52/1.53 ! [P: poly_nat,C: nat] :
% 2.52/1.53 ( ( ( synthetic_div_nat @ P @ C )
% 2.52/1.53 = zero_zero_poly_nat )
% 2.52/1.53 = ( ( degree_nat @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_67_poly__shift__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( poly_shift_real @ N @ zero_zero_poly_real )
% 2.52/1.53 = zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_68_poly__shift__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( poly_shift_poly_real @ N @ zero_z1423781445y_real )
% 2.52/1.53 = zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_69_poly__shift__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( poly_shift_nat @ N @ zero_zero_poly_nat )
% 2.52/1.53 = zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_70_order__root,axiom,
% 2.52/1.53 ! [P: poly_real,A3: real] :
% 2.52/1.53 ( ( ( poly_real2 @ P @ A3 )
% 2.52/1.53 = zero_zero_real )
% 2.52/1.53 = ( ( P = zero_zero_poly_real )
% 2.52/1.53 | ( ( order_real @ A3 @ P )
% 2.52/1.53 != zero_zero_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_71_order__root,axiom,
% 2.52/1.53 ! [P: poly_poly_real,A3: poly_real] :
% 2.52/1.53 ( ( ( poly_poly_real2 @ P @ A3 )
% 2.52/1.53 = zero_zero_poly_real )
% 2.52/1.53 = ( ( P = zero_z1423781445y_real )
% 2.52/1.53 | ( ( order_poly_real @ A3 @ P )
% 2.52/1.53 != zero_zero_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_72_order__root,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real,A3: poly_poly_real] :
% 2.52/1.53 ( ( ( poly_poly_poly_real2 @ P @ A3 )
% 2.52/1.53 = zero_z1423781445y_real )
% 2.52/1.53 = ( ( P = zero_z935034829y_real )
% 2.52/1.53 | ( ( order_poly_poly_real @ A3 @ P )
% 2.52/1.53 != zero_zero_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_73_leading__coeff__0__iff,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( ( coeff_real @ P @ ( degree_real @ P ) )
% 2.52/1.53 = zero_zero_real )
% 2.52/1.53 = ( P = zero_zero_poly_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_74_leading__coeff__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
% 2.52/1.53 = zero_zero_poly_real )
% 2.52/1.53 = ( P = zero_z1423781445y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_75_leading__coeff__0__iff,axiom,
% 2.52/1.53 ! [P: poly_nat] :
% 2.52/1.53 ( ( ( coeff_nat @ P @ ( degree_nat @ P ) )
% 2.52/1.53 = zero_zero_nat )
% 2.52/1.53 = ( P = zero_zero_poly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_76_leading__coeff__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( ( coeff_poly_poly_real @ P @ ( degree360860553y_real @ P ) )
% 2.52/1.53 = zero_z1423781445y_real )
% 2.52/1.53 = ( P = zero_z935034829y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_77_leading__coeff__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_nat] :
% 2.52/1.53 ( ( ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) )
% 2.52/1.53 = zero_zero_poly_nat )
% 2.52/1.53 = ( P = zero_z1059985641ly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_78_divide__poly__main__0,axiom,
% 2.52/1.53 ! [R2: poly_real,D: poly_real,Dr: nat,N: nat] :
% 2.52/1.53 ( ( divide1561404011n_real @ zero_zero_real @ zero_zero_poly_real @ R2 @ D @ Dr @ N )
% 2.52/1.53 = zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_79_divide__poly__main__0,axiom,
% 2.52/1.53 ! [R2: poly_poly_real,D: poly_poly_real,Dr: nat,N: nat] :
% 2.52/1.53 ( ( divide1142363123y_real @ zero_zero_poly_real @ zero_z1423781445y_real @ R2 @ D @ Dr @ N )
% 2.52/1.53 = zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_80_divide__poly__main__0,axiom,
% 2.52/1.53 ! [R2: poly_poly_poly_real,D: poly_poly_poly_real,Dr: nat,N: nat] :
% 2.52/1.53 ( ( divide924636027y_real @ zero_z1423781445y_real @ zero_z935034829y_real @ R2 @ D @ Dr @ N )
% 2.52/1.53 = zero_z935034829y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_81_zero__poly_Otransfer,axiom,
% 2.52/1.53 ( pcr_poly_real_real
% 2.52/1.53 @ ^ [Y: real,Z: real] : ( Y = Z )
% 2.52/1.53 @ ^ [Uu: nat] : zero_zero_real
% 2.52/1.53 @ zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_82_zero__poly_Otransfer,axiom,
% 2.52/1.53 ( pcr_po1314690837y_real
% 2.52/1.53 @ ^ [Y: poly_real,Z: poly_real] : ( Y = Z )
% 2.52/1.53 @ ^ [Uu: nat] : zero_zero_poly_real
% 2.52/1.53 @ zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_83_zero__poly_Otransfer,axiom,
% 2.52/1.53 ( pcr_poly_nat_nat
% 2.52/1.53 @ ^ [Y: nat,Z: nat] : ( Y = Z )
% 2.52/1.53 @ ^ [Uu: nat] : zero_zero_nat
% 2.52/1.53 @ zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_84_zero__poly_Otransfer,axiom,
% 2.52/1.53 ( pcr_po1200519205y_real
% 2.52/1.53 @ ^ [Y: poly_poly_real,Z: poly_poly_real] : ( Y = Z )
% 2.52/1.53 @ ^ [Uu: nat] : zero_z1423781445y_real
% 2.52/1.53 @ zero_z935034829y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_85_zero__poly_Otransfer,axiom,
% 2.52/1.53 ( pcr_po273983709ly_nat
% 2.52/1.53 @ ^ [Y: poly_nat,Z: poly_nat] : ( Y = Z )
% 2.52/1.53 @ ^ [Uu: nat] : zero_zero_poly_nat
% 2.52/1.53 @ zero_z1059985641ly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_86_p_I1_J,axiom,
% 2.52/1.53 ! [I2: nat] : ( member_real @ ( coeff_real @ p @ I2 ) @ ring_1_Ints_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_87_reflect__poly__0,axiom,
% 2.52/1.53 ( ( reflect_poly_real @ zero_zero_poly_real )
% 2.52/1.53 = zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_88_reflect__poly__0,axiom,
% 2.52/1.53 ( ( reflec1522834046y_real @ zero_z1423781445y_real )
% 2.52/1.53 = zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_89_reflect__poly__0,axiom,
% 2.52/1.53 ( ( reflect_poly_nat @ zero_zero_poly_nat )
% 2.52/1.53 = zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_90_synthetic__div__0,axiom,
% 2.52/1.53 ! [C: real] :
% 2.52/1.53 ( ( synthetic_div_real @ zero_zero_poly_real @ C )
% 2.52/1.53 = zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_91_synthetic__div__0,axiom,
% 2.52/1.53 ! [C: poly_real] :
% 2.52/1.53 ( ( synthe1498897281y_real @ zero_z1423781445y_real @ C )
% 2.52/1.53 = zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_92_synthetic__div__0,axiom,
% 2.52/1.53 ! [C: nat] :
% 2.52/1.53 ( ( synthetic_div_nat @ zero_zero_poly_nat @ C )
% 2.52/1.53 = zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_93_coeff__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( coeff_poly_poly_real @ zero_z935034829y_real @ N )
% 2.52/1.53 = zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_94_coeff__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( coeff_poly_nat @ zero_z1059985641ly_nat @ N )
% 2.52/1.53 = zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_95_coeff__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( coeff_real @ zero_zero_poly_real @ N )
% 2.52/1.53 = zero_zero_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_96_coeff__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( coeff_poly_real @ zero_z1423781445y_real @ N )
% 2.52/1.53 = zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_97_coeff__0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( coeff_nat @ zero_zero_poly_nat @ N )
% 2.52/1.53 = zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_98_reflect__poly__reflect__poly,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( ( coeff_real @ P @ zero_zero_nat )
% 2.52/1.53 != zero_zero_real )
% 2.52/1.53 => ( ( reflect_poly_real @ ( reflect_poly_real @ P ) )
% 2.52/1.53 = P ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_99_reflect__poly__reflect__poly,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( ( coeff_poly_real @ P @ zero_zero_nat )
% 2.52/1.53 != zero_zero_poly_real )
% 2.52/1.53 => ( ( reflec1522834046y_real @ ( reflec1522834046y_real @ P ) )
% 2.52/1.53 = P ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_100_reflect__poly__reflect__poly,axiom,
% 2.52/1.53 ! [P: poly_nat] :
% 2.52/1.53 ( ( ( coeff_nat @ P @ zero_zero_nat )
% 2.52/1.53 != zero_zero_nat )
% 2.52/1.53 => ( ( reflect_poly_nat @ ( reflect_poly_nat @ P ) )
% 2.52/1.53 = P ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_101_reflect__poly__reflect__poly,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( ( coeff_poly_poly_real @ P @ zero_zero_nat )
% 2.52/1.53 != zero_z1423781445y_real )
% 2.52/1.53 => ( ( reflec144234502y_real @ ( reflec144234502y_real @ P ) )
% 2.52/1.53 = P ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_102_reflect__poly__reflect__poly,axiom,
% 2.52/1.53 ! [P: poly_poly_nat] :
% 2.52/1.53 ( ( ( coeff_poly_nat @ P @ zero_zero_nat )
% 2.52/1.53 != zero_zero_poly_nat )
% 2.52/1.53 => ( ( reflec781175074ly_nat @ ( reflec781175074ly_nat @ P ) )
% 2.52/1.53 = P ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_103_coeff__0__reflect__poly,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( coeff_real @ ( reflect_poly_real @ P ) @ zero_zero_nat )
% 2.52/1.53 = ( coeff_real @ P @ ( degree_real @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_104_coeff__0__reflect__poly__0__iff,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( ( coeff_real @ ( reflect_poly_real @ P ) @ zero_zero_nat )
% 2.52/1.53 = zero_zero_real )
% 2.52/1.53 = ( P = zero_zero_poly_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_105_coeff__0__reflect__poly__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( ( coeff_poly_real @ ( reflec1522834046y_real @ P ) @ zero_zero_nat )
% 2.52/1.53 = zero_zero_poly_real )
% 2.52/1.53 = ( P = zero_z1423781445y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_106_coeff__0__reflect__poly__0__iff,axiom,
% 2.52/1.53 ! [P: poly_nat] :
% 2.52/1.53 ( ( ( coeff_nat @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 2.52/1.53 = zero_zero_nat )
% 2.52/1.53 = ( P = zero_zero_poly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_107_coeff__0__reflect__poly__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( ( coeff_poly_poly_real @ ( reflec144234502y_real @ P ) @ zero_zero_nat )
% 2.52/1.53 = zero_z1423781445y_real )
% 2.52/1.53 = ( P = zero_z935034829y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_108_coeff__0__reflect__poly__0__iff,axiom,
% 2.52/1.53 ! [P: poly_poly_nat] :
% 2.52/1.53 ( ( ( coeff_poly_nat @ ( reflec781175074ly_nat @ P ) @ zero_zero_nat )
% 2.52/1.53 = zero_zero_poly_nat )
% 2.52/1.53 = ( P = zero_z1059985641ly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_109_degree__reflect__poly__eq,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( ( coeff_real @ P @ zero_zero_nat )
% 2.52/1.53 != zero_zero_real )
% 2.52/1.53 => ( ( degree_real @ ( reflect_poly_real @ P ) )
% 2.52/1.53 = ( degree_real @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_110_degree__reflect__poly__eq,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( ( coeff_poly_real @ P @ zero_zero_nat )
% 2.52/1.53 != zero_zero_poly_real )
% 2.52/1.53 => ( ( degree_poly_real @ ( reflec1522834046y_real @ P ) )
% 2.52/1.53 = ( degree_poly_real @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_111_degree__reflect__poly__eq,axiom,
% 2.52/1.53 ! [P: poly_nat] :
% 2.52/1.53 ( ( ( coeff_nat @ P @ zero_zero_nat )
% 2.52/1.53 != zero_zero_nat )
% 2.52/1.53 => ( ( degree_nat @ ( reflect_poly_nat @ P ) )
% 2.52/1.53 = ( degree_nat @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_112_degree__reflect__poly__eq,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( ( coeff_poly_poly_real @ P @ zero_zero_nat )
% 2.52/1.53 != zero_z1423781445y_real )
% 2.52/1.53 => ( ( degree360860553y_real @ ( reflec144234502y_real @ P ) )
% 2.52/1.53 = ( degree360860553y_real @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_113_degree__reflect__poly__eq,axiom,
% 2.52/1.53 ! [P: poly_poly_nat] :
% 2.52/1.53 ( ( ( coeff_poly_nat @ P @ zero_zero_nat )
% 2.52/1.53 != zero_zero_poly_nat )
% 2.52/1.53 => ( ( degree_poly_nat @ ( reflec781175074ly_nat @ P ) )
% 2.52/1.53 = ( degree_poly_nat @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_114_poly__reflect__poly__0,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
% 2.52/1.53 = ( coeff_real @ P @ ( degree_real @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_115_poly__reflect__poly__0,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
% 2.52/1.53 = ( coeff_poly_real @ P @ ( degree_poly_real @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_116_poly__reflect__poly__0,axiom,
% 2.52/1.53 ! [P: poly_nat] :
% 2.52/1.53 ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
% 2.52/1.53 = ( coeff_nat @ P @ ( degree_nat @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_117_poly__reflect__poly__0,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
% 2.52/1.53 = ( coeff_poly_poly_real @ P @ ( degree360860553y_real @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_118_poly__reflect__poly__0,axiom,
% 2.52/1.53 ! [P: poly_poly_nat] :
% 2.52/1.53 ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
% 2.52/1.53 = ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_119_mem__Collect__eq,axiom,
% 2.52/1.53 ! [A3: real,P2: real > $o] :
% 2.52/1.53 ( ( member_real @ A3 @ ( collect_real @ P2 ) )
% 2.52/1.53 = ( P2 @ A3 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_120_mem__Collect__eq,axiom,
% 2.52/1.53 ! [A3: nat,P2: nat > $o] :
% 2.52/1.53 ( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
% 2.52/1.53 = ( P2 @ A3 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_121_mem__Collect__eq,axiom,
% 2.52/1.53 ! [A3: poly_real,P2: poly_real > $o] :
% 2.52/1.53 ( ( member_poly_real @ A3 @ ( collect_poly_real @ P2 ) )
% 2.52/1.53 = ( P2 @ A3 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_122_Collect__mem__eq,axiom,
% 2.52/1.53 ! [A2: set_real] :
% 2.52/1.53 ( ( collect_real
% 2.52/1.53 @ ^ [X: real] : ( member_real @ X @ A2 ) )
% 2.52/1.53 = A2 ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_123_Collect__mem__eq,axiom,
% 2.52/1.53 ! [A2: set_nat] :
% 2.52/1.53 ( ( collect_nat
% 2.52/1.53 @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
% 2.52/1.53 = A2 ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_124_Collect__mem__eq,axiom,
% 2.52/1.53 ! [A2: set_poly_real] :
% 2.52/1.53 ( ( collect_poly_real
% 2.52/1.53 @ ^ [X: poly_real] : ( member_poly_real @ X @ A2 ) )
% 2.52/1.53 = A2 ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_125_Collect__cong,axiom,
% 2.52/1.53 ! [P2: real > $o,Q: real > $o] :
% 2.52/1.53 ( ! [X3: real] :
% 2.52/1.53 ( ( P2 @ X3 )
% 2.52/1.53 = ( Q @ X3 ) )
% 2.52/1.53 => ( ( collect_real @ P2 )
% 2.52/1.53 = ( collect_real @ Q ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_126_Collect__cong,axiom,
% 2.52/1.53 ! [P2: nat > $o,Q: nat > $o] :
% 2.52/1.53 ( ! [X3: nat] :
% 2.52/1.53 ( ( P2 @ X3 )
% 2.52/1.53 = ( Q @ X3 ) )
% 2.52/1.53 => ( ( collect_nat @ P2 )
% 2.52/1.53 = ( collect_nat @ Q ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_127_Collect__cong,axiom,
% 2.52/1.53 ! [P2: poly_real > $o,Q: poly_real > $o] :
% 2.52/1.53 ( ! [X3: poly_real] :
% 2.52/1.53 ( ( P2 @ X3 )
% 2.52/1.53 = ( Q @ X3 ) )
% 2.52/1.53 => ( ( collect_poly_real @ P2 )
% 2.52/1.53 = ( collect_poly_real @ Q ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_128_poly__eqI,axiom,
% 2.52/1.53 ! [P: poly_real,Q2: poly_real] :
% 2.52/1.53 ( ! [N2: nat] :
% 2.52/1.53 ( ( coeff_real @ P @ N2 )
% 2.52/1.53 = ( coeff_real @ Q2 @ N2 ) )
% 2.52/1.53 => ( P = Q2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_129_poly__eq__iff,axiom,
% 2.52/1.53 ( ( ^ [Y: poly_real,Z: poly_real] : ( Y = Z ) )
% 2.52/1.53 = ( ^ [P3: poly_real,Q3: poly_real] :
% 2.52/1.53 ! [N3: nat] :
% 2.52/1.53 ( ( coeff_real @ P3 @ N3 )
% 2.52/1.53 = ( coeff_real @ Q3 @ N3 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_130_coeff__inject,axiom,
% 2.52/1.53 ! [X2: poly_real,Y2: poly_real] :
% 2.52/1.53 ( ( ( coeff_real @ X2 )
% 2.52/1.53 = ( coeff_real @ Y2 ) )
% 2.52/1.53 = ( X2 = Y2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_131_algebraicE,axiom,
% 2.52/1.53 ! [X2: real] :
% 2.52/1.53 ( ( algebraic_real @ X2 )
% 2.52/1.53 => ~ ! [P4: poly_real] :
% 2.52/1.53 ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
% 2.52/1.53 => ( ( P4 != zero_zero_poly_real )
% 2.52/1.53 => ( ( poly_real2 @ P4 @ X2 )
% 2.52/1.53 != zero_zero_real ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_132_algebraicI,axiom,
% 2.52/1.53 ! [P: poly_real,X2: real] :
% 2.52/1.53 ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
% 2.52/1.53 => ( ( P != zero_zero_poly_real )
% 2.52/1.53 => ( ( ( poly_real2 @ P @ X2 )
% 2.52/1.53 = zero_zero_real )
% 2.52/1.53 => ( algebraic_real @ X2 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_133_algebraic__def,axiom,
% 2.52/1.53 ( algebraic_real
% 2.52/1.53 = ( ^ [X: real] :
% 2.52/1.53 ? [P3: poly_real] :
% 2.52/1.53 ( ! [I4: nat] : ( member_real @ ( coeff_real @ P3 @ I4 ) @ ring_1_Ints_real )
% 2.52/1.53 & ( P3 != zero_zero_poly_real )
% 2.52/1.53 & ( ( poly_real2 @ P3 @ X )
% 2.52/1.53 = zero_zero_real ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_134_algebraic__altdef,axiom,
% 2.52/1.53 ( algebraic_real
% 2.52/1.53 = ( ^ [X: real] :
% 2.52/1.53 ? [P3: poly_real] :
% 2.52/1.53 ( ! [I4: nat] : ( member_real @ ( coeff_real @ P3 @ I4 ) @ field_1537545994s_real )
% 2.52/1.53 & ( P3 != zero_zero_poly_real )
% 2.52/1.53 & ( ( poly_real2 @ P3 @ X )
% 2.52/1.53 = zero_zero_real ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_135_zero__poly_Orep__eq,axiom,
% 2.52/1.53 ( ( coeff_poly_poly_real @ zero_z935034829y_real )
% 2.52/1.53 = ( ^ [Uu: nat] : zero_z1423781445y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_136_zero__poly_Orep__eq,axiom,
% 2.52/1.53 ( ( coeff_poly_nat @ zero_z1059985641ly_nat )
% 2.52/1.53 = ( ^ [Uu: nat] : zero_zero_poly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_137_zero__poly_Orep__eq,axiom,
% 2.52/1.53 ( ( coeff_real @ zero_zero_poly_real )
% 2.52/1.53 = ( ^ [Uu: nat] : zero_zero_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_138_zero__poly_Orep__eq,axiom,
% 2.52/1.53 ( ( coeff_poly_real @ zero_z1423781445y_real )
% 2.52/1.53 = ( ^ [Uu: nat] : zero_zero_poly_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_139_zero__poly_Orep__eq,axiom,
% 2.52/1.53 ( ( coeff_nat @ zero_zero_poly_nat )
% 2.52/1.53 = ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_140_poly__0__coeff__0,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( poly_real2 @ P @ zero_zero_real )
% 2.52/1.53 = ( coeff_real @ P @ zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_141_poly__0__coeff__0,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( poly_poly_real2 @ P @ zero_zero_poly_real )
% 2.52/1.53 = ( coeff_poly_real @ P @ zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_142_poly__0__coeff__0,axiom,
% 2.52/1.53 ! [P: poly_nat] :
% 2.52/1.53 ( ( poly_nat2 @ P @ zero_zero_nat )
% 2.52/1.53 = ( coeff_nat @ P @ zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_143_poly__0__coeff__0,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( poly_poly_poly_real2 @ P @ zero_z1423781445y_real )
% 2.52/1.53 = ( coeff_poly_poly_real @ P @ zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_144_poly__0__coeff__0,axiom,
% 2.52/1.53 ! [P: poly_poly_nat] :
% 2.52/1.53 ( ( poly_poly_nat2 @ P @ zero_zero_poly_nat )
% 2.52/1.53 = ( coeff_poly_nat @ P @ zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_145_order__0I,axiom,
% 2.52/1.53 ! [P: poly_real,A3: real] :
% 2.52/1.53 ( ( ( poly_real2 @ P @ A3 )
% 2.52/1.53 != zero_zero_real )
% 2.52/1.53 => ( ( order_real @ A3 @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_146_order__0I,axiom,
% 2.52/1.53 ! [P: poly_poly_real,A3: poly_real] :
% 2.52/1.53 ( ( ( poly_poly_real2 @ P @ A3 )
% 2.52/1.53 != zero_zero_poly_real )
% 2.52/1.53 => ( ( order_poly_real @ A3 @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_147_order__0I,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real,A3: poly_poly_real] :
% 2.52/1.53 ( ( ( poly_poly_poly_real2 @ P @ A3 )
% 2.52/1.53 != zero_z1423781445y_real )
% 2.52/1.53 => ( ( order_poly_poly_real @ A3 @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_148_leading__coeff__neq__0,axiom,
% 2.52/1.53 ! [P: poly_poly_poly_real] :
% 2.52/1.53 ( ( P != zero_z935034829y_real )
% 2.52/1.53 => ( ( coeff_poly_poly_real @ P @ ( degree360860553y_real @ P ) )
% 2.52/1.53 != zero_z1423781445y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_149_leading__coeff__neq__0,axiom,
% 2.52/1.53 ! [P: poly_poly_nat] :
% 2.52/1.53 ( ( P != zero_z1059985641ly_nat )
% 2.52/1.53 => ( ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) )
% 2.52/1.53 != zero_zero_poly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_150_leading__coeff__neq__0,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( P != zero_zero_poly_real )
% 2.52/1.53 => ( ( coeff_real @ P @ ( degree_real @ P ) )
% 2.52/1.53 != zero_zero_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_151_leading__coeff__neq__0,axiom,
% 2.52/1.53 ! [P: poly_poly_real] :
% 2.52/1.53 ( ( P != zero_z1423781445y_real )
% 2.52/1.53 => ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
% 2.52/1.53 != zero_zero_poly_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_152_leading__coeff__neq__0,axiom,
% 2.52/1.53 ! [P: poly_nat] :
% 2.52/1.53 ( ( P != zero_zero_poly_nat )
% 2.52/1.53 => ( ( coeff_nat @ P @ ( degree_nat @ P ) )
% 2.52/1.53 != zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_153_Rats__infinite,axiom,
% 2.52/1.53 ~ ( finite_finite_real @ field_1537545994s_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_154_Rats__0,axiom,
% 2.52/1.53 member_real @ zero_zero_real @ field_1537545994s_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_155_Ints__0,axiom,
% 2.52/1.53 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_156_Ints__0,axiom,
% 2.52/1.53 member_poly_real @ zero_zero_poly_real @ ring_1690226883y_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_157_Ints__0,axiom,
% 2.52/1.53 member1159720147y_real @ zero_z1423781445y_real @ ring_1897377867y_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_158_rsquarefree__def,axiom,
% 2.52/1.53 ( rsquarefree_real
% 2.52/1.53 = ( ^ [P3: poly_real] :
% 2.52/1.53 ( ( P3 != zero_zero_poly_real )
% 2.52/1.53 & ! [A: real] :
% 2.52/1.53 ( ( ( order_real @ A @ P3 )
% 2.52/1.53 = zero_zero_nat )
% 2.52/1.53 | ( ( order_real @ A @ P3 )
% 2.52/1.53 = one_one_nat ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_159_rsquarefree__def,axiom,
% 2.52/1.53 ( rsquar1555552848y_real
% 2.52/1.53 = ( ^ [P3: poly_poly_real] :
% 2.52/1.53 ( ( P3 != zero_z1423781445y_real )
% 2.52/1.53 & ! [A: poly_real] :
% 2.52/1.53 ( ( ( order_poly_real @ A @ P3 )
% 2.52/1.53 = zero_zero_nat )
% 2.52/1.53 | ( ( order_poly_real @ A @ P3 )
% 2.52/1.53 = one_one_nat ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_160_poly__cutoff__1,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ( N = zero_zero_nat )
% 2.52/1.53 => ( ( poly_cutoff_real @ N @ one_one_poly_real )
% 2.52/1.53 = zero_zero_poly_real ) )
% 2.52/1.53 & ( ( N != zero_zero_nat )
% 2.52/1.53 => ( ( poly_cutoff_real @ N @ one_one_poly_real )
% 2.52/1.53 = one_one_poly_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_161_poly__cutoff__1,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ( N = zero_zero_nat )
% 2.52/1.53 => ( ( poly_c1404107022y_real @ N @ one_on501200385y_real )
% 2.52/1.53 = zero_z1423781445y_real ) )
% 2.52/1.53 & ( ( N != zero_zero_nat )
% 2.52/1.53 => ( ( poly_c1404107022y_real @ N @ one_on501200385y_real )
% 2.52/1.53 = one_on501200385y_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_162_poly__cutoff__1,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ( N = zero_zero_nat )
% 2.52/1.53 => ( ( poly_cutoff_nat @ N @ one_one_poly_nat )
% 2.52/1.53 = zero_zero_poly_nat ) )
% 2.52/1.53 & ( ( N != zero_zero_nat )
% 2.52/1.53 => ( ( poly_cutoff_nat @ N @ one_one_poly_nat )
% 2.52/1.53 = one_one_poly_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_163_order__pderiv2,axiom,
% 2.52/1.53 ! [P: poly_real,A3: real,N: nat] :
% 2.52/1.53 ( ( ( pderiv_real @ P )
% 2.52/1.53 != zero_zero_poly_real )
% 2.52/1.53 => ( ( ( order_real @ A3 @ P )
% 2.52/1.53 != zero_zero_nat )
% 2.52/1.53 => ( ( ( order_real @ A3 @ ( pderiv_real @ P ) )
% 2.52/1.53 = N )
% 2.52/1.53 = ( ( order_real @ A3 @ P )
% 2.52/1.53 = ( suc @ N ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_164_order__pderiv,axiom,
% 2.52/1.53 ! [P: poly_real,A3: real] :
% 2.52/1.53 ( ( ( pderiv_real @ P )
% 2.52/1.53 != zero_zero_poly_real )
% 2.52/1.53 => ( ( ( order_real @ A3 @ P )
% 2.52/1.53 != zero_zero_nat )
% 2.52/1.53 => ( ( order_real @ A3 @ P )
% 2.52/1.53 = ( suc @ ( order_real @ A3 @ ( pderiv_real @ P ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_165_cr__poly__def,axiom,
% 2.52/1.53 ( cr_poly_real
% 2.52/1.53 = ( ^ [X: nat > real,Y3: poly_real] :
% 2.52/1.53 ( X
% 2.52/1.53 = ( coeff_real @ Y3 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_166_dvd__pderiv__iff,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( dvd_dvd_poly_real @ P @ ( pderiv_real @ P ) )
% 2.52/1.53 = ( ( degree_real @ P )
% 2.52/1.53 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_167_degree__1,axiom,
% 2.52/1.53 ( ( degree_real @ one_one_poly_real )
% 2.52/1.53 = zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_168_poly__1,axiom,
% 2.52/1.53 ! [X2: real] :
% 2.52/1.53 ( ( poly_real2 @ one_one_poly_real @ X2 )
% 2.52/1.53 = one_one_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_169_poly__1,axiom,
% 2.52/1.53 ! [X2: poly_real] :
% 2.52/1.53 ( ( poly_poly_real2 @ one_on501200385y_real @ X2 )
% 2.52/1.53 = one_one_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_170_poly__1,axiom,
% 2.52/1.53 ! [X2: nat] :
% 2.52/1.53 ( ( poly_nat2 @ one_one_poly_nat @ X2 )
% 2.52/1.53 = one_one_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_171_pderiv__1,axiom,
% 2.52/1.53 ( ( pderiv_real @ one_one_poly_real )
% 2.52/1.53 = zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_172_pderiv__1,axiom,
% 2.52/1.53 ( ( pderiv_poly_real @ one_on501200385y_real )
% 2.52/1.53 = zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_173_pderiv__1,axiom,
% 2.52/1.53 ( ( pderiv_nat @ one_one_poly_nat )
% 2.52/1.53 = zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_174_lead__coeff__1,axiom,
% 2.52/1.53 ( ( coeff_nat @ one_one_poly_nat @ ( degree_nat @ one_one_poly_nat ) )
% 2.52/1.53 = one_one_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_175_lead__coeff__1,axiom,
% 2.52/1.53 ( ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) )
% 2.52/1.53 = one_one_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_176_one__reorient,axiom,
% 2.52/1.53 ! [X2: nat] :
% 2.52/1.53 ( ( one_one_nat = X2 )
% 2.52/1.53 = ( X2 = one_one_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_177_Ints__1,axiom,
% 2.52/1.53 member_real @ one_one_real @ ring_1_Ints_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_178_Rats__1,axiom,
% 2.52/1.53 member_real @ one_one_real @ field_1537545994s_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_179_is__unit__iff__degree,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( P != zero_zero_poly_real )
% 2.52/1.53 => ( ( dvd_dvd_poly_real @ P @ one_one_poly_real )
% 2.52/1.53 = ( ( degree_real @ P )
% 2.52/1.53 = zero_zero_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_180_not__dvd__pderiv,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( ( degree_real @ P )
% 2.52/1.53 != zero_zero_nat )
% 2.52/1.53 => ~ ( dvd_dvd_poly_real @ P @ ( pderiv_real @ P ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_181_poly__shift__1,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ( N = zero_zero_nat )
% 2.52/1.53 => ( ( poly_shift_real @ N @ one_one_poly_real )
% 2.52/1.53 = one_one_poly_real ) )
% 2.52/1.53 & ( ( N != zero_zero_nat )
% 2.52/1.53 => ( ( poly_shift_real @ N @ one_one_poly_real )
% 2.52/1.53 = zero_zero_poly_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_182_poly__shift__1,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ( N = zero_zero_nat )
% 2.52/1.53 => ( ( poly_shift_poly_real @ N @ one_on501200385y_real )
% 2.52/1.53 = one_on501200385y_real ) )
% 2.52/1.53 & ( ( N != zero_zero_nat )
% 2.52/1.53 => ( ( poly_shift_poly_real @ N @ one_on501200385y_real )
% 2.52/1.53 = zero_z1423781445y_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_183_poly__shift__1,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ( N = zero_zero_nat )
% 2.52/1.53 => ( ( poly_shift_nat @ N @ one_one_poly_nat )
% 2.52/1.53 = one_one_poly_nat ) )
% 2.52/1.53 & ( ( N != zero_zero_nat )
% 2.52/1.53 => ( ( poly_shift_nat @ N @ one_one_poly_nat )
% 2.52/1.53 = zero_zero_poly_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_184_dvd__0__left__iff,axiom,
% 2.52/1.53 ! [A3: real] :
% 2.52/1.53 ( ( dvd_dvd_real @ zero_zero_real @ A3 )
% 2.52/1.53 = ( A3 = zero_zero_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_185_dvd__0__left__iff,axiom,
% 2.52/1.53 ! [A3: poly_real] :
% 2.52/1.53 ( ( dvd_dvd_poly_real @ zero_zero_poly_real @ A3 )
% 2.52/1.53 = ( A3 = zero_zero_poly_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_186_dvd__0__left__iff,axiom,
% 2.52/1.53 ! [A3: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 2.52/1.53 = ( A3 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_187_dvd__0__left__iff,axiom,
% 2.52/1.53 ! [A3: poly_poly_real] :
% 2.52/1.53 ( ( dvd_dv1946063458y_real @ zero_z1423781445y_real @ A3 )
% 2.52/1.53 = ( A3 = zero_z1423781445y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_188_dvd__0__left__iff,axiom,
% 2.52/1.53 ! [A3: poly_nat] :
% 2.52/1.53 ( ( dvd_dvd_poly_nat @ zero_zero_poly_nat @ A3 )
% 2.52/1.53 = ( A3 = zero_zero_poly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_189_dvd__0__right,axiom,
% 2.52/1.53 ! [A3: real] : ( dvd_dvd_real @ A3 @ zero_zero_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_190_dvd__0__right,axiom,
% 2.52/1.53 ! [A3: poly_real] : ( dvd_dvd_poly_real @ A3 @ zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_191_dvd__0__right,axiom,
% 2.52/1.53 ! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_192_dvd__0__right,axiom,
% 2.52/1.53 ! [A3: poly_poly_real] : ( dvd_dv1946063458y_real @ A3 @ zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_193_dvd__0__right,axiom,
% 2.52/1.53 ! [A3: poly_nat] : ( dvd_dvd_poly_nat @ A3 @ zero_zero_poly_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_194_One__nat__def,axiom,
% 2.52/1.53 ( one_one_nat
% 2.52/1.53 = ( suc @ zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_195_not__is__unit__0,axiom,
% 2.52/1.53 ~ ( dvd_dvd_poly_real @ zero_zero_poly_real @ one_one_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_196_not__is__unit__0,axiom,
% 2.52/1.53 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_197_not__is__unit__0,axiom,
% 2.52/1.53 ~ ( dvd_dv1946063458y_real @ zero_z1423781445y_real @ one_on501200385y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_198_nat_Oinject,axiom,
% 2.52/1.53 ! [X22: nat,Y22: nat] :
% 2.52/1.53 ( ( ( suc @ X22 )
% 2.52/1.53 = ( suc @ Y22 ) )
% 2.52/1.53 = ( X22 = Y22 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_199_old_Onat_Oinject,axiom,
% 2.52/1.53 ! [Nat: nat,Nat2: nat] :
% 2.52/1.53 ( ( ( suc @ Nat )
% 2.52/1.53 = ( suc @ Nat2 ) )
% 2.52/1.53 = ( Nat = Nat2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_200_nat__dvd__1__iff__1,axiom,
% 2.52/1.53 ! [M: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 2.52/1.53 = ( M = one_one_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_201_dvd__1__iff__1,axiom,
% 2.52/1.53 ! [M: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 2.52/1.53 = ( M
% 2.52/1.53 = ( suc @ zero_zero_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_202_dvd__1__left,axiom,
% 2.52/1.53 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_203_dvd__trans,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat,C: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 => ( ( dvd_dvd_nat @ B2 @ C )
% 2.52/1.53 => ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_204_dvd__refl,axiom,
% 2.52/1.53 ! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_205_n__not__Suc__n,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( N
% 2.52/1.53 != ( suc @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_206_Suc__inject,axiom,
% 2.52/1.53 ! [X2: nat,Y2: nat] :
% 2.52/1.53 ( ( ( suc @ X2 )
% 2.52/1.53 = ( suc @ Y2 ) )
% 2.52/1.53 => ( X2 = Y2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_207_zero__neq__one,axiom,
% 2.52/1.53 zero_zero_real != one_one_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_208_zero__neq__one,axiom,
% 2.52/1.53 zero_zero_poly_real != one_one_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_209_zero__neq__one,axiom,
% 2.52/1.53 zero_zero_nat != one_one_nat ).
% 2.52/1.53
% 2.52/1.53 thf(fact_210_zero__neq__one,axiom,
% 2.52/1.53 zero_z1423781445y_real != one_on501200385y_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_211_zero__neq__one,axiom,
% 2.52/1.53 zero_zero_poly_nat != one_one_poly_nat ).
% 2.52/1.53
% 2.52/1.53 thf(fact_212_dvd__0__left,axiom,
% 2.52/1.53 ! [A3: real] :
% 2.52/1.53 ( ( dvd_dvd_real @ zero_zero_real @ A3 )
% 2.52/1.53 => ( A3 = zero_zero_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_213_dvd__0__left,axiom,
% 2.52/1.53 ! [A3: poly_real] :
% 2.52/1.53 ( ( dvd_dvd_poly_real @ zero_zero_poly_real @ A3 )
% 2.52/1.53 => ( A3 = zero_zero_poly_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_214_dvd__0__left,axiom,
% 2.52/1.53 ! [A3: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 2.52/1.53 => ( A3 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_215_dvd__0__left,axiom,
% 2.52/1.53 ! [A3: poly_poly_real] :
% 2.52/1.53 ( ( dvd_dv1946063458y_real @ zero_z1423781445y_real @ A3 )
% 2.52/1.53 => ( A3 = zero_z1423781445y_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_216_dvd__0__left,axiom,
% 2.52/1.53 ! [A3: poly_nat] :
% 2.52/1.53 ( ( dvd_dvd_poly_nat @ zero_zero_poly_nat @ A3 )
% 2.52/1.53 => ( A3 = zero_zero_poly_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_217_dvd__unit__imp__unit,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 2.52/1.53 => ( dvd_dvd_nat @ A3 @ one_one_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_218_unit__imp__dvd,axiom,
% 2.52/1.53 ! [B2: nat,A3: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 2.52/1.53 => ( dvd_dvd_nat @ B2 @ A3 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_219_one__dvd,axiom,
% 2.52/1.53 ! [A3: nat] : ( dvd_dvd_nat @ one_one_nat @ A3 ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_220_not0__implies__Suc,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( N != zero_zero_nat )
% 2.52/1.53 => ? [M2: nat] :
% 2.52/1.53 ( N
% 2.52/1.53 = ( suc @ M2 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_221_old_Onat_Oinducts,axiom,
% 2.52/1.53 ! [P2: nat > $o,Nat: nat] :
% 2.52/1.53 ( ( P2 @ zero_zero_nat )
% 2.52/1.53 => ( ! [Nat3: nat] :
% 2.52/1.53 ( ( P2 @ Nat3 )
% 2.52/1.53 => ( P2 @ ( suc @ Nat3 ) ) )
% 2.52/1.53 => ( P2 @ Nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_222_old_Onat_Oexhaust,axiom,
% 2.52/1.53 ! [Y2: nat] :
% 2.52/1.53 ( ( Y2 != zero_zero_nat )
% 2.52/1.53 => ~ ! [Nat3: nat] :
% 2.52/1.53 ( Y2
% 2.52/1.53 != ( suc @ Nat3 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_223_Zero__not__Suc,axiom,
% 2.52/1.53 ! [M: nat] :
% 2.52/1.53 ( zero_zero_nat
% 2.52/1.53 != ( suc @ M ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_224_Zero__neq__Suc,axiom,
% 2.52/1.53 ! [M: nat] :
% 2.52/1.53 ( zero_zero_nat
% 2.52/1.53 != ( suc @ M ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_225_Suc__neq__Zero,axiom,
% 2.52/1.53 ! [M: nat] :
% 2.52/1.53 ( ( suc @ M )
% 2.52/1.53 != zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_226_zero__induct,axiom,
% 2.52/1.53 ! [P2: nat > $o,K: nat] :
% 2.52/1.53 ( ( P2 @ K )
% 2.52/1.53 => ( ! [N2: nat] :
% 2.52/1.53 ( ( P2 @ ( suc @ N2 ) )
% 2.52/1.53 => ( P2 @ N2 ) )
% 2.52/1.53 => ( P2 @ zero_zero_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_227_diff__induct,axiom,
% 2.52/1.53 ! [P2: nat > nat > $o,M: nat,N: nat] :
% 2.52/1.53 ( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
% 2.52/1.53 => ( ! [Y4: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y4 ) )
% 2.52/1.53 => ( ! [X3: nat,Y4: nat] :
% 2.52/1.53 ( ( P2 @ X3 @ Y4 )
% 2.52/1.53 => ( P2 @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 2.52/1.53 => ( P2 @ M @ N ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_228_nat__induct,axiom,
% 2.52/1.53 ! [P2: nat > $o,N: nat] :
% 2.52/1.53 ( ( P2 @ zero_zero_nat )
% 2.52/1.53 => ( ! [N2: nat] :
% 2.52/1.53 ( ( P2 @ N2 )
% 2.52/1.53 => ( P2 @ ( suc @ N2 ) ) )
% 2.52/1.53 => ( P2 @ N ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_229_nat_OdiscI,axiom,
% 2.52/1.53 ! [Nat: nat,X22: nat] :
% 2.52/1.53 ( ( Nat
% 2.52/1.53 = ( suc @ X22 ) )
% 2.52/1.53 => ( Nat != zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_230_old_Onat_Odistinct_I1_J,axiom,
% 2.52/1.53 ! [Nat2: nat] :
% 2.52/1.53 ( zero_zero_nat
% 2.52/1.53 != ( suc @ Nat2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_231_old_Onat_Odistinct_I2_J,axiom,
% 2.52/1.53 ! [Nat2: nat] :
% 2.52/1.53 ( ( suc @ Nat2 )
% 2.52/1.53 != zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_232_nat_Odistinct_I1_J,axiom,
% 2.52/1.53 ! [X22: nat] :
% 2.52/1.53 ( zero_zero_nat
% 2.52/1.53 != ( suc @ X22 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_233_algebraic__int_Oinducts,axiom,
% 2.52/1.53 ! [X2: real,P2: real > $o] :
% 2.52/1.53 ( ( algebraic_int_real @ X2 )
% 2.52/1.53 => ( ! [P4: poly_real,X3: real] :
% 2.52/1.53 ( ( ( coeff_real @ P4 @ ( degree_real @ P4 ) )
% 2.52/1.53 = one_one_real )
% 2.52/1.53 => ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
% 2.52/1.53 => ( ( ( poly_real2 @ P4 @ X3 )
% 2.52/1.53 = zero_zero_real )
% 2.52/1.53 => ( P2 @ X3 ) ) ) )
% 2.52/1.53 => ( P2 @ X2 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_234_algebraic__int_Ointros,axiom,
% 2.52/1.53 ! [P: poly_real,X2: real] :
% 2.52/1.53 ( ( ( coeff_real @ P @ ( degree_real @ P ) )
% 2.52/1.53 = one_one_real )
% 2.52/1.53 => ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
% 2.52/1.53 => ( ( ( poly_real2 @ P @ X2 )
% 2.52/1.53 = zero_zero_real )
% 2.52/1.53 => ( algebraic_int_real @ X2 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_235_algebraic__int_Osimps,axiom,
% 2.52/1.53 ( algebraic_int_real
% 2.52/1.53 = ( ^ [A: real] :
% 2.52/1.53 ? [P3: poly_real,X: real] :
% 2.52/1.53 ( ( A = X )
% 2.52/1.53 & ( ( coeff_real @ P3 @ ( degree_real @ P3 ) )
% 2.52/1.53 = one_one_real )
% 2.52/1.53 & ! [I4: nat] : ( member_real @ ( coeff_real @ P3 @ I4 ) @ ring_1_Ints_real )
% 2.52/1.53 & ( ( poly_real2 @ P3 @ X )
% 2.52/1.53 = zero_zero_real ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_236_algebraic__int_Ocases,axiom,
% 2.52/1.53 ! [A3: real] :
% 2.52/1.53 ( ( algebraic_int_real @ A3 )
% 2.52/1.53 => ~ ! [P4: poly_real] :
% 2.52/1.53 ( ( ( coeff_real @ P4 @ ( degree_real @ P4 ) )
% 2.52/1.53 = one_one_real )
% 2.52/1.53 => ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
% 2.52/1.53 => ( ( poly_real2 @ P4 @ A3 )
% 2.52/1.53 != zero_zero_real ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_237_exists__least__lemma,axiom,
% 2.52/1.53 ! [P2: nat > $o] :
% 2.52/1.53 ( ~ ( P2 @ zero_zero_nat )
% 2.52/1.53 => ( ? [X_12: nat] : ( P2 @ X_12 )
% 2.52/1.53 => ? [N2: nat] :
% 2.52/1.53 ( ~ ( P2 @ N2 )
% 2.52/1.53 & ( P2 @ ( suc @ N2 ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_238_algebraic__int__0,axiom,
% 2.52/1.53 algebraic_int_real @ zero_zero_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_239_dvd__antisym,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ M @ N )
% 2.52/1.53 => ( ( dvd_dvd_nat @ N @ M )
% 2.52/1.53 => ( M = N ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_240_int__imp__algebraic__int,axiom,
% 2.52/1.53 ! [X2: real] :
% 2.52/1.53 ( ( member_real @ X2 @ ring_1_Ints_real )
% 2.52/1.53 => ( algebraic_int_real @ X2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_241_algebraic__int__imp__algebraic,axiom,
% 2.52/1.53 ! [X2: real] :
% 2.52/1.53 ( ( algebraic_int_real @ X2 )
% 2.52/1.53 => ( algebraic_real @ X2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_242_rational__algebraic__int__is__int,axiom,
% 2.52/1.53 ! [X2: real] :
% 2.52/1.53 ( ( algebraic_int_real @ X2 )
% 2.52/1.53 => ( ( member_real @ X2 @ field_1537545994s_real )
% 2.52/1.53 => ( member_real @ X2 @ ring_1_Ints_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_243_algebraic__int__inverse,axiom,
% 2.52/1.53 ! [P: poly_real,X2: real] :
% 2.52/1.53 ( ( ( poly_real2 @ P @ X2 )
% 2.52/1.53 = zero_zero_real )
% 2.52/1.53 => ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
% 2.52/1.53 => ( ( ( coeff_real @ P @ zero_zero_nat )
% 2.52/1.53 = one_one_real )
% 2.52/1.53 => ( algebraic_int_real @ ( inverse_inverse_real @ X2 ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_244_algebraic__int__root,axiom,
% 2.52/1.53 ! [Y2: real,P: poly_real,X2: real] :
% 2.52/1.53 ( ( algebraic_int_real @ Y2 )
% 2.52/1.53 => ( ( ( poly_real2 @ P @ X2 )
% 2.52/1.53 = Y2 )
% 2.52/1.53 => ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
% 2.52/1.53 => ( ( ( coeff_real @ P @ ( degree_real @ P ) )
% 2.52/1.53 = one_one_real )
% 2.52/1.53 => ( ( ord_less_nat @ zero_zero_nat @ ( degree_real @ P ) )
% 2.52/1.53 => ( algebraic_int_real @ X2 ) ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_245_gcd__nat_Oextremum__uniqueI,axiom,
% 2.52/1.53 ! [A3: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 2.52/1.53 => ( A3 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_246_gcd__nat_Onot__eq__extremum,axiom,
% 2.52/1.53 ! [A3: nat] :
% 2.52/1.53 ( ( A3 != zero_zero_nat )
% 2.52/1.53 = ( ( dvd_dvd_nat @ A3 @ zero_zero_nat )
% 2.52/1.53 & ( A3 != zero_zero_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_247_finite__Collect__less__nat,axiom,
% 2.52/1.53 ! [K: nat] :
% 2.52/1.53 ( finite_finite_nat
% 2.52/1.53 @ ( collect_nat
% 2.52/1.53 @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_248_not__gr__zero,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.52/1.53 = ( N = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_249_bot__nat__0_Onot__eq__extremum,axiom,
% 2.52/1.53 ! [A3: nat] :
% 2.52/1.53 ( ( A3 != zero_zero_nat )
% 2.52/1.53 = ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_250_less__nat__zero__code,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_251_neq0__conv,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( N != zero_zero_nat )
% 2.52/1.53 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_252_lessI,axiom,
% 2.52/1.53 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_253_Suc__mono,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M @ N )
% 2.52/1.53 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_254_Suc__less__eq,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.52/1.53 = ( ord_less_nat @ M @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_255_zero__less__Suc,axiom,
% 2.52/1.53 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_256_less__Suc0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 2.52/1.53 = ( N = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_257_less__one,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ N @ one_one_nat )
% 2.52/1.53 = ( N = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_258_zero__less__iff__neq__zero,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.52/1.53 = ( N != zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_259_gr__implies__not__zero,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M @ N )
% 2.52/1.53 => ( N != zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_260_not__less__zero,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_261_gr__zeroI,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( N != zero_zero_nat )
% 2.52/1.53 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_262_gr0I,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( N != zero_zero_nat )
% 2.52/1.53 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_263_not__gr0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.52/1.53 = ( N = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_264_not__less0,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_265_less__zeroE,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_266_gr__implies__not0,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M @ N )
% 2.52/1.53 => ( N != zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_267_infinite__descent0,axiom,
% 2.52/1.53 ! [P2: nat > $o,N: nat] :
% 2.52/1.53 ( ( P2 @ zero_zero_nat )
% 2.52/1.53 => ( ! [N2: nat] :
% 2.52/1.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.52/1.53 => ( ~ ( P2 @ N2 )
% 2.52/1.53 => ? [M3: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M3 @ N2 )
% 2.52/1.53 & ~ ( P2 @ M3 ) ) ) )
% 2.52/1.53 => ( P2 @ N ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_268_bot__nat__0_Oextremum__strict,axiom,
% 2.52/1.53 ! [A3: nat] :
% 2.52/1.53 ~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_269_not__less__less__Suc__eq,axiom,
% 2.52/1.53 ! [N: nat,M: nat] :
% 2.52/1.53 ( ~ ( ord_less_nat @ N @ M )
% 2.52/1.53 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 2.52/1.53 = ( N = M ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_270_strict__inc__induct,axiom,
% 2.52/1.53 ! [I2: nat,J: nat,P2: nat > $o] :
% 2.52/1.53 ( ( ord_less_nat @ I2 @ J )
% 2.52/1.53 => ( ! [I3: nat] :
% 2.52/1.53 ( ( J
% 2.52/1.53 = ( suc @ I3 ) )
% 2.52/1.53 => ( P2 @ I3 ) )
% 2.52/1.53 => ( ! [I3: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I3 @ J )
% 2.52/1.53 => ( ( P2 @ ( suc @ I3 ) )
% 2.52/1.53 => ( P2 @ I3 ) ) )
% 2.52/1.53 => ( P2 @ I2 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_271_less__Suc__induct,axiom,
% 2.52/1.53 ! [I2: nat,J: nat,P2: nat > nat > $o] :
% 2.52/1.53 ( ( ord_less_nat @ I2 @ J )
% 2.52/1.53 => ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
% 2.52/1.53 => ( ! [I3: nat,J2: nat,K2: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I3 @ J2 )
% 2.52/1.53 => ( ( ord_less_nat @ J2 @ K2 )
% 2.52/1.53 => ( ( P2 @ I3 @ J2 )
% 2.52/1.53 => ( ( P2 @ J2 @ K2 )
% 2.52/1.53 => ( P2 @ I3 @ K2 ) ) ) ) )
% 2.52/1.53 => ( P2 @ I2 @ J ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_272_less__trans__Suc,axiom,
% 2.52/1.53 ! [I2: nat,J: nat,K: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I2 @ J )
% 2.52/1.53 => ( ( ord_less_nat @ J @ K )
% 2.52/1.53 => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_273_Suc__less__SucD,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.52/1.53 => ( ord_less_nat @ M @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_274_less__antisym,axiom,
% 2.52/1.53 ! [N: nat,M: nat] :
% 2.52/1.53 ( ~ ( ord_less_nat @ N @ M )
% 2.52/1.53 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 2.52/1.53 => ( M = N ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_275_Suc__less__eq2,axiom,
% 2.52/1.53 ! [N: nat,M: nat] :
% 2.52/1.53 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 2.52/1.53 = ( ? [M4: nat] :
% 2.52/1.53 ( ( M
% 2.52/1.53 = ( suc @ M4 ) )
% 2.52/1.53 & ( ord_less_nat @ N @ M4 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_276_All__less__Suc,axiom,
% 2.52/1.53 ! [N: nat,P2: nat > $o] :
% 2.52/1.53 ( ( ! [I4: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 2.52/1.53 => ( P2 @ I4 ) ) )
% 2.52/1.53 = ( ( P2 @ N )
% 2.52/1.53 & ! [I4: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I4 @ N )
% 2.52/1.53 => ( P2 @ I4 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_277_not__less__eq,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ~ ( ord_less_nat @ M @ N ) )
% 2.52/1.53 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_278_less__Suc__eq,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.52/1.53 = ( ( ord_less_nat @ M @ N )
% 2.52/1.53 | ( M = N ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_279_Ex__less__Suc,axiom,
% 2.52/1.53 ! [N: nat,P2: nat > $o] :
% 2.52/1.53 ( ( ? [I4: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 2.52/1.53 & ( P2 @ I4 ) ) )
% 2.52/1.53 = ( ( P2 @ N )
% 2.52/1.53 | ? [I4: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I4 @ N )
% 2.52/1.53 & ( P2 @ I4 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_280_less__SucI,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M @ N )
% 2.52/1.53 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_281_less__SucE,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.52/1.53 => ( ~ ( ord_less_nat @ M @ N )
% 2.52/1.53 => ( M = N ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_282_Suc__lessI,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M @ N )
% 2.52/1.53 => ( ( ( suc @ M )
% 2.52/1.53 != N )
% 2.52/1.53 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_283_Suc__lessE,axiom,
% 2.52/1.53 ! [I2: nat,K: nat] :
% 2.52/1.53 ( ( ord_less_nat @ ( suc @ I2 ) @ K )
% 2.52/1.53 => ~ ! [J2: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I2 @ J2 )
% 2.52/1.53 => ( K
% 2.52/1.53 != ( suc @ J2 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_284_Suc__lessD,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 2.52/1.53 => ( ord_less_nat @ M @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_285_Nat_OlessE,axiom,
% 2.52/1.53 ! [I2: nat,K: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I2 @ K )
% 2.52/1.53 => ( ( K
% 2.52/1.53 != ( suc @ I2 ) )
% 2.52/1.53 => ~ ! [J2: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I2 @ J2 )
% 2.52/1.53 => ( K
% 2.52/1.53 != ( suc @ J2 ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_286_Rats__inverse,axiom,
% 2.52/1.53 ! [A3: real] :
% 2.52/1.53 ( ( member_real @ A3 @ field_1537545994s_real )
% 2.52/1.53 => ( member_real @ ( inverse_inverse_real @ A3 ) @ field_1537545994s_real ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_287_nat__neq__iff,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( M != N )
% 2.52/1.53 = ( ( ord_less_nat @ M @ N )
% 2.52/1.53 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_288_less__not__refl,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ~ ( ord_less_nat @ N @ N ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_289_less__not__refl2,axiom,
% 2.52/1.53 ! [N: nat,M: nat] :
% 2.52/1.53 ( ( ord_less_nat @ N @ M )
% 2.52/1.53 => ( M != N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_290_less__not__refl3,axiom,
% 2.52/1.53 ! [S: nat,T: nat] :
% 2.52/1.53 ( ( ord_less_nat @ S @ T )
% 2.52/1.53 => ( S != T ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_291_less__irrefl__nat,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ~ ( ord_less_nat @ N @ N ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_292_nat__less__induct,axiom,
% 2.52/1.53 ! [P2: nat > $o,N: nat] :
% 2.52/1.53 ( ! [N2: nat] :
% 2.52/1.53 ( ! [M3: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M3 @ N2 )
% 2.52/1.53 => ( P2 @ M3 ) )
% 2.52/1.53 => ( P2 @ N2 ) )
% 2.52/1.53 => ( P2 @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_293_infinite__descent,axiom,
% 2.52/1.53 ! [P2: nat > $o,N: nat] :
% 2.52/1.53 ( ! [N2: nat] :
% 2.52/1.53 ( ~ ( P2 @ N2 )
% 2.52/1.53 => ? [M3: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M3 @ N2 )
% 2.52/1.53 & ~ ( P2 @ M3 ) ) )
% 2.52/1.53 => ( P2 @ N ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_294_linorder__neqE__nat,axiom,
% 2.52/1.53 ! [X2: nat,Y2: nat] :
% 2.52/1.53 ( ( X2 != Y2 )
% 2.52/1.53 => ( ~ ( ord_less_nat @ X2 @ Y2 )
% 2.52/1.53 => ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_295_linorder__neqE__linordered__idom,axiom,
% 2.52/1.53 ! [X2: real,Y2: real] :
% 2.52/1.53 ( ( X2 != Y2 )
% 2.52/1.53 => ( ~ ( ord_less_real @ X2 @ Y2 )
% 2.52/1.53 => ( ord_less_real @ Y2 @ X2 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_296_lift__Suc__mono__less__iff,axiom,
% 2.52/1.53 ! [F: nat > nat,N: nat,M: nat] :
% 2.52/1.53 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.52/1.53 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 2.52/1.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_297_lift__Suc__mono__less__iff,axiom,
% 2.52/1.53 ! [F: nat > real,N: nat,M: nat] :
% 2.52/1.53 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.52/1.53 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 2.52/1.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_298_lift__Suc__mono__less,axiom,
% 2.52/1.53 ! [F: nat > nat,N: nat,N4: nat] :
% 2.52/1.53 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.52/1.53 => ( ( ord_less_nat @ N @ N4 )
% 2.52/1.53 => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_299_lift__Suc__mono__less,axiom,
% 2.52/1.53 ! [F: nat > real,N: nat,N4: nat] :
% 2.52/1.53 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.52/1.53 => ( ( ord_less_nat @ N @ N4 )
% 2.52/1.53 => ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_300_dvd__pos__nat,axiom,
% 2.52/1.53 ! [N: nat,M: nat] :
% 2.52/1.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.52/1.53 => ( ( dvd_dvd_nat @ M @ N )
% 2.52/1.53 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_301_finite__divisors__nat,axiom,
% 2.52/1.53 ! [M: nat] :
% 2.52/1.53 ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.52/1.53 => ( finite_finite_nat
% 2.52/1.53 @ ( collect_nat
% 2.52/1.53 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_302_not__one__less__zero,axiom,
% 2.52/1.53 ~ ( ord_less_poly_real @ one_one_poly_real @ zero_zero_poly_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_303_not__one__less__zero,axiom,
% 2.52/1.53 ~ ( ord_le38482960y_real @ one_on501200385y_real @ zero_z1423781445y_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_304_not__one__less__zero,axiom,
% 2.52/1.53 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_305_not__one__less__zero,axiom,
% 2.52/1.53 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_306_zero__less__one,axiom,
% 2.52/1.53 ord_less_poly_real @ zero_zero_poly_real @ one_one_poly_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_307_zero__less__one,axiom,
% 2.52/1.53 ord_le38482960y_real @ zero_z1423781445y_real @ one_on501200385y_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_308_zero__less__one,axiom,
% 2.52/1.53 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 2.52/1.53
% 2.52/1.53 thf(fact_309_zero__less__one,axiom,
% 2.52/1.53 ord_less_real @ zero_zero_real @ one_one_real ).
% 2.52/1.53
% 2.52/1.53 thf(fact_310_less__Suc__eq__0__disj,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.52/1.53 = ( ( M = zero_zero_nat )
% 2.52/1.53 | ? [J3: nat] :
% 2.52/1.53 ( ( M
% 2.52/1.53 = ( suc @ J3 ) )
% 2.52/1.53 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_311_gr0__implies__Suc,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.52/1.53 => ? [M2: nat] :
% 2.52/1.53 ( N
% 2.52/1.53 = ( suc @ M2 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_312_All__less__Suc2,axiom,
% 2.52/1.53 ! [N: nat,P2: nat > $o] :
% 2.52/1.53 ( ( ! [I4: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 2.52/1.53 => ( P2 @ I4 ) ) )
% 2.52/1.53 = ( ( P2 @ zero_zero_nat )
% 2.52/1.53 & ! [I4: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I4 @ N )
% 2.52/1.53 => ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_313_gr0__conv__Suc,axiom,
% 2.52/1.53 ! [N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.52/1.53 = ( ? [M5: nat] :
% 2.52/1.53 ( N
% 2.52/1.53 = ( suc @ M5 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_314_Ex__less__Suc2,axiom,
% 2.52/1.53 ! [N: nat,P2: nat > $o] :
% 2.52/1.53 ( ( ? [I4: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 2.52/1.53 & ( P2 @ I4 ) ) )
% 2.52/1.53 = ( ( P2 @ zero_zero_nat )
% 2.52/1.53 | ? [I4: nat] :
% 2.52/1.53 ( ( ord_less_nat @ I4 @ N )
% 2.52/1.53 & ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_315_nat__dvd__not__less,axiom,
% 2.52/1.53 ! [M: nat,N: nat] :
% 2.52/1.53 ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.52/1.53 => ( ( ord_less_nat @ M @ N )
% 2.52/1.53 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_316_nat__induct__non__zero,axiom,
% 2.52/1.53 ! [N: nat,P2: nat > $o] :
% 2.52/1.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.52/1.53 => ( ( P2 @ one_one_nat )
% 2.52/1.53 => ( ! [N2: nat] :
% 2.52/1.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.52/1.53 => ( ( P2 @ N2 )
% 2.52/1.53 => ( P2 @ ( suc @ N2 ) ) ) )
% 2.52/1.53 => ( P2 @ N ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_317_less__degree__imp,axiom,
% 2.52/1.53 ! [N: nat,P: poly_poly_real] :
% 2.52/1.53 ( ( ord_less_nat @ N @ ( degree_poly_real @ P ) )
% 2.52/1.53 => ? [I3: nat] :
% 2.52/1.53 ( ( ord_less_nat @ N @ I3 )
% 2.52/1.53 & ( ( coeff_poly_real @ P @ I3 )
% 2.52/1.53 != zero_zero_poly_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_318_less__degree__imp,axiom,
% 2.52/1.53 ! [N: nat,P: poly_nat] :
% 2.52/1.53 ( ( ord_less_nat @ N @ ( degree_nat @ P ) )
% 2.52/1.53 => ? [I3: nat] :
% 2.52/1.53 ( ( ord_less_nat @ N @ I3 )
% 2.52/1.53 & ( ( coeff_nat @ P @ I3 )
% 2.52/1.53 != zero_zero_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_319_less__degree__imp,axiom,
% 2.52/1.53 ! [N: nat,P: poly_poly_poly_real] :
% 2.52/1.53 ( ( ord_less_nat @ N @ ( degree360860553y_real @ P ) )
% 2.52/1.53 => ? [I3: nat] :
% 2.52/1.53 ( ( ord_less_nat @ N @ I3 )
% 2.52/1.53 & ( ( coeff_poly_poly_real @ P @ I3 )
% 2.52/1.53 != zero_z1423781445y_real ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_320_less__degree__imp,axiom,
% 2.52/1.53 ! [N: nat,P: poly_poly_nat] :
% 2.52/1.53 ( ( ord_less_nat @ N @ ( degree_poly_nat @ P ) )
% 2.52/1.53 => ? [I3: nat] :
% 2.52/1.53 ( ( ord_less_nat @ N @ I3 )
% 2.52/1.53 & ( ( coeff_poly_nat @ P @ I3 )
% 2.52/1.53 != zero_zero_poly_nat ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_321_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat] :
% 2.52/1.53 ( ( A3 != B2 )
% 2.52/1.53 => ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 => ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 & ( A3 != B2 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_322_gcd__nat_Ostrict__implies__not__eq,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat] :
% 2.52/1.53 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 & ( A3 != B2 ) )
% 2.52/1.53 => ( A3 != B2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_323_gcd__nat_Ostrict__implies__order,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat] :
% 2.52/1.53 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 & ( A3 != B2 ) )
% 2.52/1.53 => ( dvd_dvd_nat @ A3 @ B2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_324_gcd__nat_Ostrict__iff__order,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat] :
% 2.52/1.53 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 & ( A3 != B2 ) )
% 2.52/1.53 = ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 & ( A3 != B2 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_325_gcd__nat_Oorder__iff__strict,axiom,
% 2.52/1.53 ( dvd_dvd_nat
% 2.52/1.53 = ( ^ [A: nat,B3: nat] :
% 2.52/1.53 ( ( ( dvd_dvd_nat @ A @ B3 )
% 2.52/1.53 & ( A != B3 ) )
% 2.52/1.53 | ( A = B3 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_326_gcd__nat_Ostrict__trans2,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat,C: nat] :
% 2.52/1.53 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 & ( A3 != B2 ) )
% 2.52/1.53 => ( ( dvd_dvd_nat @ B2 @ C )
% 2.52/1.53 => ( ( dvd_dvd_nat @ A3 @ C )
% 2.52/1.53 & ( A3 != C ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_327_gcd__nat_Ostrict__trans1,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat,C: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 => ( ( ( dvd_dvd_nat @ B2 @ C )
% 2.52/1.53 & ( B2 != C ) )
% 2.52/1.53 => ( ( dvd_dvd_nat @ A3 @ C )
% 2.52/1.53 & ( A3 != C ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_328_gcd__nat_Ostrict__trans,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat,C: nat] :
% 2.52/1.53 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 & ( A3 != B2 ) )
% 2.52/1.53 => ( ( ( dvd_dvd_nat @ B2 @ C )
% 2.52/1.53 & ( B2 != C ) )
% 2.52/1.53 => ( ( dvd_dvd_nat @ A3 @ C )
% 2.52/1.53 & ( A3 != C ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_329_gcd__nat_Oantisym,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 => ( ( dvd_dvd_nat @ B2 @ A3 )
% 2.52/1.53 => ( A3 = B2 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_330_gcd__nat_Oirrefl,axiom,
% 2.52/1.53 ! [A3: nat] :
% 2.52/1.53 ~ ( ( dvd_dvd_nat @ A3 @ A3 )
% 2.52/1.53 & ( A3 != A3 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_331_gcd__nat_Oeq__iff,axiom,
% 2.52/1.53 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 2.52/1.53 = ( ^ [A: nat,B3: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ A @ B3 )
% 2.52/1.53 & ( dvd_dvd_nat @ B3 @ A ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_332_gcd__nat_Otrans,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat,C: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 => ( ( dvd_dvd_nat @ B2 @ C )
% 2.52/1.53 => ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_333_gcd__nat_Orefl,axiom,
% 2.52/1.53 ! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_334_gcd__nat_Oasym,axiom,
% 2.52/1.53 ! [A3: nat,B2: nat] :
% 2.52/1.53 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 2.52/1.53 & ( A3 != B2 ) )
% 2.52/1.53 => ~ ( ( dvd_dvd_nat @ B2 @ A3 )
% 2.52/1.53 & ( B2 != A3 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_335_gcd__nat_Oextremum,axiom,
% 2.52/1.53 ! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_336_gcd__nat_Oextremum__strict,axiom,
% 2.52/1.53 ! [A3: nat] :
% 2.52/1.53 ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 2.52/1.53 & ( zero_zero_nat != A3 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_337_gcd__nat_Oextremum__unique,axiom,
% 2.52/1.53 ! [A3: nat] :
% 2.52/1.53 ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
% 2.52/1.53 = ( A3 = zero_zero_nat ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_338_poly__IVT__pos,axiom,
% 2.52/1.53 ! [A3: real,B2: real,P: poly_real] :
% 2.52/1.53 ( ( ord_less_real @ A3 @ B2 )
% 2.52/1.53 => ( ( ord_less_real @ ( poly_real2 @ P @ A3 ) @ zero_zero_real )
% 2.52/1.53 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B2 ) )
% 2.52/1.53 => ? [X3: real] :
% 2.52/1.53 ( ( ord_less_real @ A3 @ X3 )
% 2.52/1.53 & ( ord_less_real @ X3 @ B2 )
% 2.52/1.53 & ( ( poly_real2 @ P @ X3 )
% 2.52/1.53 = zero_zero_real ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_339_poly__IVT__neg,axiom,
% 2.52/1.53 ! [A3: real,B2: real,P: poly_real] :
% 2.52/1.53 ( ( ord_less_real @ A3 @ B2 )
% 2.52/1.53 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A3 ) )
% 2.52/1.53 => ( ( ord_less_real @ ( poly_real2 @ P @ B2 ) @ zero_zero_real )
% 2.52/1.53 => ? [X3: real] :
% 2.52/1.53 ( ( ord_less_real @ A3 @ X3 )
% 2.52/1.53 & ( ord_less_real @ X3 @ B2 )
% 2.52/1.53 & ( ( poly_real2 @ P @ X3 )
% 2.52/1.53 = zero_zero_real ) ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_340_finite__M__bounded__by__nat,axiom,
% 2.52/1.53 ! [P2: nat > $o,I2: nat] :
% 2.52/1.53 ( finite_finite_nat
% 2.52/1.53 @ ( collect_nat
% 2.52/1.53 @ ^ [K3: nat] :
% 2.52/1.53 ( ( P2 @ K3 )
% 2.52/1.53 & ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_341_Rats__dense__in__real,axiom,
% 2.52/1.53 ! [X2: real,Y2: real] :
% 2.52/1.53 ( ( ord_less_real @ X2 @ Y2 )
% 2.52/1.53 => ? [X3: real] :
% 2.52/1.53 ( ( member_real @ X3 @ field_1537545994s_real )
% 2.52/1.53 & ( ord_less_real @ X2 @ X3 )
% 2.52/1.53 & ( ord_less_real @ X3 @ Y2 ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_342_Rats__no__bot__less,axiom,
% 2.52/1.53 ! [X2: real] :
% 2.52/1.53 ? [X3: real] :
% 2.52/1.53 ( ( member_real @ X3 @ field_1537545994s_real )
% 2.52/1.53 & ( ord_less_real @ X3 @ X2 ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(fact_343_poly__pinfty__gt__lc,axiom,
% 2.52/1.53 ! [P: poly_real] :
% 2.52/1.53 ( ( ord_less_real @ zero_zero_real @ ( coeff_real @ P @ ( degree_real @ P ) ) )
% 2.52/1.53 => ? [N2: real] :
% 2.52/1.53 ! [X4: real] :
% 2.52/1.53 ( ( ord_less_eq_real @ N2 @ X4 )
% 2.52/1.53 => ( ord_less_eq_real @ ( coeff_real @ P @ ( degree_real @ P ) ) @ ( poly_real2 @ P @ X4 ) ) ) ) ).
% 2.52/1.53
% 2.52/1.53 thf(conj_0,conjecture,
% 2.52/1.53 ( finite_finite_real
% 2.52/1.53 @ ( collect_real
% 2.52/1.53 @ ^ [X: real] :
% 2.52/1.53 ( ( poly_real2 @ ( pderiv_real @ p ) @ X )
% 2.52/1.53 = zero_zero_real ) ) ) ).
% 2.52/1.53 ------------------------
% 2.52/1.55 % (11746)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/183Mi)
% 2.52/1.55 % (11747)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_11513 for (2999ds/4Mi)
% 2.52/1.55 % (11750)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_11513 for (2999ds/2Mi)
% 2.52/1.55 % (11750)Instruction limit reached!
% 2.52/1.55 % (11750)------------------------------
% 2.52/1.55 % (11750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.55 % (11750)Termination reason: Unknown
% 2.52/1.55 % (11750)Termination phase: shuffling
% 2.52/1.55
% 2.52/1.55 % (11750)Memory used [KB]: 1407
% 2.52/1.55 % (11750)Time elapsed: 0.002 s
% 2.52/1.55 % (11750)Instructions burned: 3 (million)
% 2.52/1.55 % (11750)------------------------------
% 2.52/1.55 % (11750)------------------------------
% 2.52/1.55 % (11752)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/18Mi)
% 2.52/1.55 % (11747)Instruction limit reached!
% 2.52/1.55 % (11747)------------------------------
% 2.52/1.55 % (11747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.55 % (11747)Termination reason: Unknown
% 2.52/1.55 % (11747)Termination phase: shuffling
% 2.52/1.55
% 2.52/1.55 % (11747)Memory used [KB]: 1407
% 2.52/1.55 % (11747)Time elapsed: 0.003 s
% 2.52/1.55 % (11747)Instructions burned: 6 (million)
% 2.52/1.55 % (11747)------------------------------
% 2.52/1.55 % (11747)------------------------------
% 2.52/1.56 % (11748)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_11513 for (2999ds/27Mi)
% 2.52/1.56 % (11749)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/2Mi)
% 2.52/1.56 % (11751)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_11513 for (2999ds/275Mi)
% 2.52/1.56 % (11752)Instruction limit reached!
% 2.52/1.56 % (11752)------------------------------
% 2.52/1.56 % (11752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.56 % (11752)Termination reason: Unknown
% 2.52/1.56 % (11752)Termination phase: Property scanning
% 2.52/1.56
% 2.52/1.56 % (11752)Memory used [KB]: 1663
% 2.52/1.56 % (11752)Time elapsed: 0.010 s
% 2.52/1.56 % (11752)Instructions burned: 19 (million)
% 2.52/1.56 % (11752)------------------------------
% 2.52/1.56 % (11752)------------------------------
% 2.52/1.56 % (11749)Instruction limit reached!
% 2.52/1.56 % (11749)------------------------------
% 2.52/1.56 % (11749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.56 % (11749)Termination reason: Unknown
% 2.52/1.56 % (11749)Termination phase: shuffling
% 2.52/1.56
% 2.52/1.56 % (11749)Memory used [KB]: 1407
% 2.52/1.56 % (11749)Time elapsed: 0.004 s
% 2.52/1.56 % (11749)Instructions burned: 3 (million)
% 2.52/1.56 % (11749)------------------------------
% 2.52/1.56 % (11749)------------------------------
% 2.52/1.56 % (11758)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/3Mi)
% 2.52/1.56 % (11760)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_11513 for (2999ds/37Mi)
% 2.52/1.56 % (11758)Instruction limit reached!
% 2.52/1.56 % (11758)------------------------------
% 2.52/1.56 % (11758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.56 % (11758)Termination reason: Unknown
% 2.52/1.56 % (11758)Termination phase: shuffling
% 2.52/1.56
% 2.52/1.56 % (11758)Memory used [KB]: 1407
% 2.52/1.56 % (11758)Time elapsed: 0.002 s
% 2.52/1.56 % (11758)Instructions burned: 3 (million)
% 2.52/1.56 % (11758)------------------------------
% 2.52/1.56 % (11758)------------------------------
% 2.52/1.57 % (11763)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_11513 for (2999ds/3Mi)
% 2.52/1.57 % (11763)Instruction limit reached!
% 2.52/1.57 % (11763)------------------------------
% 2.52/1.57 % (11763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (11763)Termination reason: Unknown
% 2.52/1.57 % (11763)Termination phase: shuffling
% 2.52/1.57
% 2.52/1.57 % (11763)Memory used [KB]: 1407
% 2.52/1.57 % (11763)Time elapsed: 0.002 s
% 2.52/1.57 % (11763)Instructions burned: 3 (million)
% 2.52/1.57 % (11763)------------------------------
% 2.52/1.57 % (11763)------------------------------
% 2.52/1.57 % (11764)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_11513 for (2999ds/1041Mi)
% 2.52/1.57 % (11762)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_11513 for (2999ds/15Mi)
% 2.52/1.57 % (11760)Instruction limit reached!
% 2.52/1.57 % (11760)------------------------------
% 2.52/1.57 % (11760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (11760)Termination reason: Unknown
% 2.52/1.57 % (11760)Termination phase: Property scanning
% 2.52/1.57
% 2.52/1.57 % (11760)Memory used [KB]: 2046
% 2.52/1.57 % (11760)Time elapsed: 0.012 s
% 2.52/1.57 % (11760)Instructions burned: 40 (million)
% 2.52/1.57 % (11760)------------------------------
% 2.52/1.57 % (11760)------------------------------
% 2.52/1.57 % (11748)Instruction limit reached!
% 2.52/1.57 % (11748)------------------------------
% 2.52/1.57 % (11748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (11748)Termination reason: Unknown
% 2.52/1.57 % (11748)Termination phase: Property scanning
% 2.52/1.57
% 2.52/1.57 % (11748)Memory used [KB]: 1663
% 2.52/1.57 % (11748)Time elapsed: 0.018 s
% 2.52/1.57 % (11748)Instructions burned: 28 (million)
% 2.52/1.57 % (11748)------------------------------
% 2.52/1.57 % (11748)------------------------------
% 2.52/1.58 % (11762)Instruction limit reached!
% 2.52/1.58 % (11762)------------------------------
% 2.52/1.58 % (11762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.58 % (11762)Termination reason: Unknown
% 2.52/1.58 % (11762)Termination phase: Property scanning
% 2.52/1.58
% 2.52/1.58 % (11762)Memory used [KB]: 1663
% 2.52/1.58 % (11762)Time elapsed: 0.007 s
% 2.52/1.58 % (11762)Instructions burned: 18 (million)
% 2.52/1.58 % (11762)------------------------------
% 2.52/1.58 % (11762)------------------------------
% 2.52/1.58 % (11770)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/7Mi)
% 2.52/1.58 % (11771)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/16Mi)
% 2.52/1.58 % (11770)Instruction limit reached!
% 2.52/1.58 % (11770)------------------------------
% 2.52/1.58 % (11770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.58 % (11770)Termination reason: Unknown
% 2.52/1.58 % (11770)Termination phase: shuffling
% 2.52/1.58
% 2.52/1.58 % (11770)Memory used [KB]: 1535
% 2.52/1.58 % (11770)Time elapsed: 0.004 s
% 2.52/1.58 % (11770)Instructions burned: 7 (million)
% 2.52/1.58 % (11770)------------------------------
% 2.52/1.58 % (11770)------------------------------
% 2.52/1.59 % (11774)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_11513 for (2999ds/3Mi)
% 2.52/1.59 % (11771)Instruction limit reached!
% 2.52/1.59 % (11771)------------------------------
% 2.52/1.59 % (11771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (11771)Termination reason: Unknown
% 2.52/1.59 % (11771)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (11771)Memory used [KB]: 1663
% 2.52/1.59 % (11771)Time elapsed: 0.006 s
% 2.52/1.59 % (11771)Instructions burned: 16 (million)
% 2.52/1.59 % (11771)------------------------------
% 2.52/1.59 % (11771)------------------------------
% 2.52/1.59 % (11774)Instruction limit reached!
% 2.52/1.59 % (11774)------------------------------
% 2.52/1.59 % (11774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (11774)Termination reason: Unknown
% 2.52/1.59 % (11774)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (11774)Memory used [KB]: 1407
% 2.52/1.59 % (11774)Time elapsed: 0.002 s
% 2.52/1.59 % (11774)Instructions burned: 3 (million)
% 2.52/1.59 % (11774)------------------------------
% 2.52/1.59 % (11774)------------------------------
% 2.52/1.59 % (11751)Refutation not found, incomplete strategy
% 2.52/1.59 % (11751)------------------------------
% 2.52/1.59 % (11751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (11751)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.59
% 2.52/1.59
% 2.52/1.59 % (11751)Memory used [KB]: 6396
% 2.52/1.59 % (11751)Time elapsed: 0.033 s
% 2.52/1.59 % (11751)Instructions burned: 52 (million)
% 2.52/1.59 % (11751)------------------------------
% 2.52/1.59 % (11751)------------------------------
% 2.52/1.59 % (11773)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_11513 for (2999ds/3Mi)
% 2.52/1.59 % (11773)Instruction limit reached!
% 2.52/1.59 % (11773)------------------------------
% 2.52/1.59 % (11773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (11773)Termination reason: Unknown
% 2.52/1.59 % (11773)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (11773)Memory used [KB]: 1407
% 2.52/1.59 % (11773)Time elapsed: 0.003 s
% 2.52/1.59 % (11773)Instructions burned: 3 (million)
% 2.52/1.59 % (11773)------------------------------
% 2.52/1.59 % (11773)------------------------------
% 2.52/1.60 % (11779)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/3Mi)
% 2.52/1.60 % (11780)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/4Mi)
% 2.52/1.60 % (11779)Instruction limit reached!
% 2.52/1.60 % (11779)------------------------------
% 2.52/1.60 % (11779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.60 % (11779)Termination reason: Unknown
% 2.52/1.60 % (11779)Termination phase: shuffling
% 2.52/1.60
% 2.52/1.60 % (11779)Memory used [KB]: 1407
% 2.52/1.60 % (11779)Time elapsed: 0.002 s
% 2.52/1.60 % (11779)Instructions burned: 3 (million)
% 2.52/1.60 % (11779)------------------------------
% 2.52/1.60 % (11779)------------------------------
% 2.58/1.60 % (11780)Instruction limit reached!
% 2.58/1.60 % (11780)------------------------------
% 2.58/1.60 % (11780)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.60 % (11780)Termination reason: Unknown
% 2.58/1.60 % (11780)Termination phase: shuffling
% 2.58/1.60
% 2.58/1.60 % (11780)Memory used [KB]: 1407
% 2.58/1.60 % (11780)Time elapsed: 0.003 s
% 2.58/1.60 % (11780)Instructions burned: 6 (million)
% 2.58/1.60 % (11780)------------------------------
% 2.58/1.60 % (11780)------------------------------
% 2.58/1.60 % (11776)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/7Mi)
% 2.58/1.60 % (11746)Instruction limit reached!
% 2.58/1.60 % (11746)------------------------------
% 2.58/1.60 % (11746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.60 % (11746)Termination reason: Unknown
% 2.58/1.60 % (11746)Termination phase: Saturation
% 2.58/1.60
% 2.58/1.60 % (11746)Memory used [KB]: 7291
% 2.58/1.60 % (11746)Time elapsed: 0.055 s
% 2.58/1.60 % (11746)Instructions burned: 184 (million)
% 2.58/1.60 % (11746)------------------------------
% 2.58/1.60 % (11746)------------------------------
% 2.58/1.60 % (11776)Instruction limit reached!
% 2.58/1.60 % (11776)------------------------------
% 2.58/1.60 % (11776)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.60 % (11776)Termination reason: Unknown
% 2.58/1.60 % (11776)Termination phase: shuffling
% 2.58/1.60
% 2.58/1.60 % (11776)Memory used [KB]: 1535
% 2.58/1.60 % (11776)Time elapsed: 0.007 s
% 2.58/1.60 % (11776)Instructions burned: 8 (million)
% 2.58/1.60 % (11776)------------------------------
% 2.58/1.60 % (11776)------------------------------
% 2.58/1.61 % (11787)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/6Mi)
% 2.58/1.61 % (11784)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/710Mi)
% 2.58/1.61 % (11788)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_11513 for (2999ds/902Mi)
% 2.58/1.61 % (11787)Instruction limit reached!
% 2.58/1.61 % (11787)------------------------------
% 2.58/1.61 % (11787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.61 % (11787)Termination reason: Unknown
% 2.58/1.61 % (11787)Termination phase: shuffling
% 2.58/1.61
% 2.58/1.61 % (11787)Memory used [KB]: 1407
% 2.58/1.61 % (11787)Time elapsed: 0.003 s
% 2.58/1.61 % (11787)Instructions burned: 6 (million)
% 2.58/1.61 % (11787)------------------------------
% 2.58/1.61 % (11787)------------------------------
% 2.58/1.61 % (11782)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/18Mi)
% 2.58/1.61 % (11792)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_11513 for (2999ds/21Mi)
% 2.58/1.62 % (11796)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_11513 for (2999ds/6Mi)
% 2.58/1.62 % (11796)Instruction limit reached!
% 2.58/1.62 % (11796)------------------------------
% 2.58/1.62 % (11796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.62 % (11796)Termination reason: Unknown
% 2.58/1.62 % (11796)Termination phase: shuffling
% 2.58/1.62
% 2.58/1.62 % (11796)Memory used [KB]: 1407
% 2.58/1.62 % (11796)Time elapsed: 0.003 s
% 2.58/1.62 % (11796)Instructions burned: 6 (million)
% 2.58/1.62 % (11796)------------------------------
% 2.58/1.62 % (11796)------------------------------
% 2.58/1.62 % (11792)Instruction limit reached!
% 2.58/1.62 % (11792)------------------------------
% 2.58/1.62 % (11792)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.62 % (11792)Termination reason: Unknown
% 2.58/1.62 % (11792)Termination phase: Property scanning
% 2.58/1.62
% 2.58/1.62 % (11792)Memory used [KB]: 1663
% 2.58/1.62 % (11792)Time elapsed: 0.008 s
% 2.58/1.62 % (11792)Instructions burned: 21 (million)
% 2.58/1.62 % (11792)------------------------------
% 2.58/1.62 % (11792)------------------------------
% 2.58/1.62 % (11782)Instruction limit reached!
% 2.58/1.62 % (11782)------------------------------
% 2.58/1.62 % (11782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.62 % (11782)Termination reason: Unknown
% 2.58/1.62 % (11782)Termination phase: Property scanning
% 2.58/1.62
% 2.58/1.62 % (11782)Memory used [KB]: 1663
% 2.58/1.62 % (11782)Time elapsed: 0.013 s
% 2.58/1.62 % (11782)Instructions burned: 18 (million)
% 2.58/1.62 % (11782)------------------------------
% 2.58/1.62 % (11782)------------------------------
% 2.58/1.62 % (11793)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/5Mi)
% 2.58/1.63 % (11793)Instruction limit reached!
% 2.58/1.63 % (11793)------------------------------
% 2.58/1.63 % (11793)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.63 % (11793)Termination reason: Unknown
% 2.58/1.63 % (11793)Termination phase: shuffling
% 2.58/1.63
% 2.58/1.63 % (11793)Memory used [KB]: 1407
% 2.58/1.63 % (11793)Time elapsed: 0.005 s
% 2.58/1.63 % (11793)Instructions burned: 6 (million)
% 2.58/1.63 % (11793)------------------------------
% 2.58/1.63 % (11793)------------------------------
% 2.58/1.63 % (11801)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/377Mi)
% 2.58/1.63 % (11803)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/779Mi)
% 2.58/1.64 % (11804)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/19Mi)
% 2.58/1.64 % (11807)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_11513 for (2999ds/879Mi)
% 2.58/1.65 % (11804)Instruction limit reached!
% 2.58/1.65 % (11804)------------------------------
% 2.58/1.65 % (11804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.65 % (11804)Termination reason: Unknown
% 2.58/1.65 % (11804)Termination phase: shuffling
% 2.58/1.65
% 2.58/1.65 % (11804)Memory used [KB]: 1791
% 2.58/1.65 % (11804)Time elapsed: 0.012 s
% 2.58/1.65 % (11804)Instructions burned: 20 (million)
% 2.58/1.65 % (11804)------------------------------
% 2.58/1.65 % (11804)------------------------------
% 2.58/1.66 % (11814)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_11513 for (2998ds/17Mi)
% 2.58/1.66 % (11814)Instruction limit reached!
% 2.58/1.66 % (11814)------------------------------
% 2.58/1.66 % (11814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.66 % (11814)Termination reason: Unknown
% 2.58/1.66 % (11814)Termination phase: Property scanning
% 2.58/1.66
% 2.58/1.66 % (11814)Memory used [KB]: 1663
% 2.58/1.66 % (11814)Time elapsed: 0.007 s
% 2.58/1.66 % (11814)Instructions burned: 19 (million)
% 2.58/1.66 % (11814)------------------------------
% 2.58/1.66 % (11814)------------------------------
% 2.58/1.67 % (11817)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_11513 for (2998ds/3Mi)
% 2.58/1.67 % (11817)Instruction limit reached!
% 2.58/1.67 % (11817)------------------------------
% 2.58/1.67 % (11817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.67 % (11817)Termination reason: Unknown
% 2.58/1.67 % (11817)Termination phase: shuffling
% 2.58/1.67
% 2.58/1.67 % (11817)Memory used [KB]: 1407
% 2.58/1.67 % (11817)Time elapsed: 0.002 s
% 2.58/1.67 % (11817)Instructions burned: 3 (million)
% 2.58/1.67 % (11817)------------------------------
% 2.58/1.67 % (11817)------------------------------
% 2.58/1.68 % (11820)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_11513 for (2998ds/30Mi)
% 2.58/1.69 % (11820)Instruction limit reached!
% 2.58/1.69 % (11820)------------------------------
% 2.58/1.69 % (11820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.69 % (11820)Termination reason: Unknown
% 2.58/1.69 % (11820)Termination phase: Preprocessing 1
% 2.58/1.69
% 2.58/1.69 % (11820)Memory used [KB]: 1791
% 2.58/1.69 % (11820)Time elapsed: 0.011 s
% 2.58/1.69 % (11820)Instructions burned: 33 (million)
% 2.58/1.69 % (11820)------------------------------
% 2.58/1.69 % (11820)------------------------------
% 2.58/1.70 % (11827)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_11513 for (2998ds/127Mi)
% 2.58/1.74 % (11827)Instruction limit reached!
% 2.58/1.74 % (11827)------------------------------
% 2.58/1.74 % (11827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.74 % (11827)Termination reason: Unknown
% 2.58/1.74 % (11827)Termination phase: Saturation
% 2.58/1.74
% 2.58/1.74 % (11827)Memory used [KB]: 7036
% 2.58/1.74 % (11827)Time elapsed: 0.037 s
% 2.58/1.74 % (11827)Instructions burned: 129 (million)
% 2.58/1.74 % (11827)------------------------------
% 2.58/1.74 % (11827)------------------------------
% 2.58/1.74 % (11801)Instruction limit reached!
% 2.58/1.74 % (11801)------------------------------
% 2.58/1.74 % (11801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.74 % (11801)Termination reason: Unknown
% 2.58/1.74 % (11801)Termination phase: Saturation
% 2.58/1.74
% 2.58/1.74 % (11801)Memory used [KB]: 7931
% 2.58/1.74 % (11801)Time elapsed: 0.117 s
% 2.58/1.74 % (11801)Instructions burned: 379 (million)
% 2.58/1.74 % (11801)------------------------------
% 2.58/1.74 % (11801)------------------------------
% 2.58/1.75 % (11837)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_11513 for (2998ds/100Mi)
% 2.58/1.75 % (11839)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_11513 for (2998ds/3Mi)
% 2.58/1.76 % (11839)Instruction limit reached!
% 2.58/1.76 % (11839)------------------------------
% 2.58/1.76 % (11839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.76 % (11839)Termination reason: Unknown
% 2.58/1.76 % (11839)Termination phase: shuffling
% 2.58/1.76
% 2.58/1.76 % (11839)Memory used [KB]: 1407
% 2.58/1.76 % (11839)Time elapsed: 0.003 s
% 2.58/1.76 % (11839)Instructions burned: 6 (million)
% 2.58/1.76 % (11839)------------------------------
% 2.58/1.76 % (11839)------------------------------
% 2.58/1.77 % (11847)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_11513 for (2997ds/20Mi)
% 2.58/1.77 % (11847)Instruction limit reached!
% 2.58/1.77 % (11847)------------------------------
% 2.58/1.77 % (11847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.77 % (11847)Termination reason: Unknown
% 2.58/1.77 % (11847)Termination phase: Property scanning
% 2.58/1.77
% 2.58/1.77 % (11847)Memory used [KB]: 1663
% 2.58/1.77 % (11847)Time elapsed: 0.008 s
% 2.58/1.77 % (11847)Instructions burned: 23 (million)
% 2.58/1.77 % (11847)------------------------------
% 2.58/1.77 % (11847)------------------------------
% 2.58/1.77 % (11837)Instruction limit reached!
% 2.58/1.77 % (11837)------------------------------
% 2.58/1.77 % (11837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.77 % (11837)Termination reason: Unknown
% 2.58/1.77 % (11837)Termination phase: Property scanning
% 2.58/1.77
% 2.58/1.77 % (11837)Memory used [KB]: 2302
% 2.58/1.77 % (11837)Time elapsed: 0.028 s
% 2.58/1.77 % (11837)Instructions burned: 103 (million)
% 2.58/1.77 % (11837)------------------------------
% 2.58/1.77 % (11837)------------------------------
% 2.58/1.78 % (11857)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_11513 for (2997ds/86Mi)
% 2.58/1.78 % (11860)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_11513 for (2997ds/107Mi)
% 2.58/1.81 % (11857)Instruction limit reached!
% 2.58/1.81 % (11857)------------------------------
% 2.58/1.81 % (11857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.81 % (11857)Termination reason: Unknown
% 2.58/1.81 % (11857)Termination phase: Saturation
% 2.58/1.81
% 2.58/1.81 % (11857)Memory used [KB]: 6652
% 2.58/1.81 % (11857)Time elapsed: 0.025 s
% 2.58/1.81 % (11857)Instructions burned: 87 (million)
% 2.58/1.81 % (11857)------------------------------
% 2.58/1.81 % (11857)------------------------------
% 2.58/1.81 % (11877)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_11513 for (2997ds/39Mi)
% 2.58/1.81 % (11860)Instruction limit reached!
% 2.58/1.81 % (11860)------------------------------
% 2.58/1.81 % (11860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.82 % (11860)Termination reason: Unknown
% 2.58/1.82 % (11860)Termination phase: Saturation
% 2.58/1.82
% 2.58/1.82 % (11860)Memory used [KB]: 6908
% 2.58/1.82 % (11860)Time elapsed: 0.032 s
% 2.58/1.82 % (11860)Instructions burned: 109 (million)
% 2.58/1.82 % (11860)------------------------------
% 2.58/1.82 % (11860)------------------------------
% 2.58/1.82 % (11784)Instruction limit reached!
% 2.58/1.82 % (11784)------------------------------
% 2.58/1.82 % (11784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.82 % (11784)Termination reason: Unknown
% 2.58/1.82 % (11784)Termination phase: Saturation
% 2.58/1.82
% 2.58/1.82 % (11784)Memory used [KB]: 7803
% 2.58/1.82 % (11784)Time elapsed: 0.218 s
% 2.58/1.82 % (11784)Instructions burned: 711 (million)
% 2.58/1.82 % (11784)------------------------------
% 2.58/1.82 % (11784)------------------------------
% 2.58/1.82 % (11881)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_11513 for (2997ds/448Mi)
% 2.58/1.83 % (11877)Instruction limit reached!
% 2.58/1.83 % (11877)------------------------------
% 2.58/1.83 % (11877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.83 % (11877)Termination reason: Unknown
% 2.58/1.83 % (11877)Termination phase: Property scanning
% 2.58/1.83
% 2.58/1.83 % (11877)Memory used [KB]: 2046
% 2.58/1.83 % (11877)Time elapsed: 0.012 s
% 2.58/1.83 % (11877)Instructions burned: 40 (million)
% 2.58/1.83 % (11877)------------------------------
% 2.58/1.83 % (11877)------------------------------
% 2.58/1.83 % (11883)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_11513 for (2997ds/46Mi)
% 2.58/1.84 % (11884)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_11513 for (2997ds/98Mi)
% 2.58/1.85 % (11883)Instruction limit reached!
% 2.58/1.85 % (11883)------------------------------
% 2.58/1.85 % (11883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.85 % (11883)Termination reason: Unknown
% 2.58/1.85 % (11883)Termination phase: Property scanning
% 2.58/1.85
% 2.58/1.85 % (11883)Memory used [KB]: 2046
% 2.58/1.85 % (11883)Time elapsed: 0.015 s
% 2.58/1.85 % (11883)Instructions burned: 48 (million)
% 2.58/1.85 % (11883)------------------------------
% 2.58/1.85 % (11883)------------------------------
% 2.58/1.86 % (11884)Instruction limit reached!
% 2.58/1.86 % (11884)------------------------------
% 2.58/1.86 % (11884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.86 % (11884)Termination reason: Unknown
% 2.58/1.86 % (11884)Termination phase: Property scanning
% 2.58/1.86
% 2.58/1.86 % (11884)Memory used [KB]: 2302
% 2.58/1.86 % (11884)Time elapsed: 0.027 s
% 2.58/1.86 % (11884)Instructions burned: 101 (million)
% 2.58/1.86 % (11884)------------------------------
% 2.58/1.86 % (11884)------------------------------
% 2.58/1.86 % (11803)Instruction limit reached!
% 2.58/1.86 % (11803)------------------------------
% 2.58/1.86 % (11803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.86 % (11803)Termination reason: Unknown
% 2.58/1.86 % (11803)Termination phase: Saturation
% 2.58/1.86
% 2.58/1.86 % (11803)Memory used [KB]: 7931
% 2.58/1.86 % (11803)Time elapsed: 0.235 s
% 2.58/1.86 % (11803)Instructions burned: 779 (million)
% 2.58/1.86 % (11803)------------------------------
% 2.58/1.86 % (11803)------------------------------
% 2.58/1.86 % (11885)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/507Mi)
% 2.58/1.87 % (11886)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/149Mi)
% 2.58/1.87 % (11887)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/39Mi)
% 2.58/1.87 % (11788)Instruction limit reached!
% 2.58/1.87 % (11788)------------------------------
% 2.58/1.87 % (11788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.87 % (11788)Termination reason: Unknown
% 2.58/1.87 % (11788)Termination phase: Saturation
% 2.58/1.87
% 2.58/1.87 % (11788)Memory used [KB]: 8571
% 2.58/1.87 % (11788)Time elapsed: 0.267 s
% 2.58/1.87 % (11788)Instructions burned: 902 (million)
% 2.58/1.87 % (11788)------------------------------
% 2.58/1.87 % (11788)------------------------------
% 2.58/1.88 % (11888)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/6Mi)
% 2.58/1.88 % (11888)Instruction limit reached!
% 2.58/1.88 % (11888)------------------------------
% 2.58/1.88 % (11888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.88 % (11888)Termination reason: Unknown
% 2.58/1.88 % (11888)Termination phase: shuffling
% 2.58/1.88
% 2.58/1.88 % (11888)Memory used [KB]: 1407
% 2.58/1.88 % (11888)Time elapsed: 0.003 s
% 2.58/1.88 % (11888)Instructions burned: 7 (million)
% 2.58/1.88 % (11888)------------------------------
% 2.58/1.88 % (11888)------------------------------
% 2.58/1.88 % (11887)Instruction limit reached!
% 2.58/1.88 % (11887)------------------------------
% 2.58/1.88 % (11887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.88 % (11887)Termination reason: Unknown
% 2.58/1.88 % (11887)Termination phase: Property scanning
% 2.58/1.88
% 2.58/1.88 % (11887)Memory used [KB]: 2046
% 2.58/1.88 % (11887)Time elapsed: 0.013 s
% 2.58/1.88 % (11887)Instructions burned: 40 (million)
% 2.58/1.88 % (11887)------------------------------
% 2.58/1.88 % (11887)------------------------------
% 2.58/1.89 % (11764)Instruction limit reached!
% 2.58/1.89 % (11764)------------------------------
% 2.58/1.89 % (11764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.89 % (11764)Termination reason: Unknown
% 2.58/1.89 % (11764)Termination phase: Saturation
% 2.58/1.89
% 2.58/1.89 % (11764)Memory used [KB]: 9722
% 2.58/1.89 % (11764)Time elapsed: 0.322 s
% 2.58/1.89 % (11764)Instructions burned: 1043 (million)
% 2.58/1.89 % (11764)------------------------------
% 2.58/1.89 % (11764)------------------------------
% 2.58/1.89 % (11889)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/1087Mi)
% 2.58/1.89 % (11890)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/47Mi)
% 2.58/1.90 % (11891)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/44Mi)
% 2.58/1.91 % (11890)Instruction limit reached!
% 2.58/1.91 % (11890)------------------------------
% 2.58/1.91 % (11890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.91 % (11890)Termination reason: Unknown
% 2.58/1.91 % (11890)Termination phase: Property scanning
% 2.58/1.91
% 2.58/1.91 % (11890)Memory used [KB]: 2046
% 2.58/1.91 % (11890)Time elapsed: 0.016 s
% 2.58/1.91 % (11890)Instructions burned: 50 (million)
% 2.58/1.91 % (11890)------------------------------
% 2.58/1.91 % (11890)------------------------------
% 2.58/1.91 % (11891)Instruction limit reached!
% 2.58/1.91 % (11891)------------------------------
% 2.58/1.91 % (11891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.91 % (11891)Termination reason: Unknown
% 2.58/1.91 % (11891)Termination phase: Property scanning
% 2.58/1.91
% 2.58/1.91 % (11891)Memory used [KB]: 2174
% 2.58/1.91 % (11891)Time elapsed: 0.014 s
% 2.58/1.91 % (11891)Instructions burned: 47 (million)
% 2.58/1.91 % (11891)------------------------------
% 2.58/1.91 % (11891)------------------------------
% 2.58/1.92 % (11886)Instruction limit reached!
% 2.58/1.92 % (11886)------------------------------
% 2.58/1.92 % (11886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.92 % (11886)Termination reason: Unknown
% 2.58/1.92 % (11886)Termination phase: Saturation
% 2.58/1.92
% 2.58/1.92 % (11886)Memory used [KB]: 7291
% 2.58/1.92 % (11886)Time elapsed: 0.046 s
% 2.58/1.92 % (11886)Instructions burned: 151 (million)
% 2.58/1.92 % (11886)------------------------------
% 2.58/1.92 % (11886)------------------------------
% 2.58/1.92 % (11892)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/432Mi)
% 2.58/1.92 % (11893)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/62Mi)
% 2.58/1.93 % (11894)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_11513 for (2996ds/31Mi)
% 2.58/1.93 % (11881)First to succeed.
% 2.58/1.93 % (11807)Instruction limit reached!
% 2.58/1.93 % (11807)------------------------------
% 2.58/1.93 % (11807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.93 % (11807)Termination reason: Unknown
% 2.58/1.93 % (11807)Termination phase: Saturation
% 2.58/1.93
% 2.58/1.93 % (11807)Memory used [KB]: 8059
% 2.58/1.93 % (11807)Time elapsed: 0.292 s
% 2.58/1.93 % (11807)Instructions burned: 879 (million)
% 2.58/1.93 % (11807)------------------------------
% 2.58/1.93 % (11807)------------------------------
% 2.58/1.93 % (11881)Refutation found. Thanks to Tanya!
% 2.58/1.93 % SZS status Theorem for DTF2THF_11513
% 2.58/1.93 % SZS output start Proof for DTF2THF_11513
% 2.58/1.93 thf(type_def_5, type, real: $tType).
% 2.58/1.93 thf(type_def_7, type, set_nat: $tType).
% 2.58/1.93 thf(type_def_8, type, set_poly_poly_real: $tType).
% 2.58/1.93 thf(type_def_9, type, set_poly_real: $tType).
% 2.58/1.93 thf(type_def_10, type, set_real: $tType).
% 2.58/1.93 thf(type_def_11, type, nat: $tType).
% 2.58/1.93 thf(type_def_12, type, poly_nat: $tType).
% 2.58/1.93 thf(type_def_13, type, poly_poly_real: $tType).
% 2.58/1.93 thf(type_def_14, type, poly_real: $tType).
% 2.58/1.93 thf(type_def_15, type, poly_poly_nat: $tType).
% 2.58/1.93 thf(type_def_16, type, poly_poly_poly_real: $tType).
% 2.58/1.93 thf(func_def_0, type, poly_poly_poly_real: $tType).
% 2.58/1.93 thf(func_def_1, type, set_poly_poly_real: $tType).
% 2.58/1.93 thf(func_def_2, type, poly_poly_real: $tType).
% 2.58/1.93 thf(func_def_3, type, poly_poly_nat: $tType).
% 2.58/1.93 thf(func_def_4, type, set_poly_real: $tType).
% 2.58/1.93 thf(func_def_5, type, poly_real: $tType).
% 2.58/1.93 thf(func_def_6, type, poly_nat: $tType).
% 2.58/1.93 thf(func_def_7, type, set_real: $tType).
% 2.58/1.93 thf(func_def_8, type, set_nat: $tType).
% 2.58/1.93 thf(func_def_9, type, real: $tType).
% 2.58/1.93 thf(func_def_10, type, nat: $tType).
% 2.58/1.93 thf(func_def_11, type, inverse_inverse_real: real > real).
% 2.58/1.93 thf(func_def_12, type, finite_finite_nat: set_nat > $o).
% 2.58/1.93 thf(func_def_13, type, finite1328464339y_real: set_poly_poly_real > $o).
% 2.58/1.93 thf(func_def_14, type, finite1810960971y_real: set_poly_real > $o).
% 2.58/1.93 thf(func_def_15, type, finite_finite_real: set_real > $o).
% 2.58/1.93 thf(func_def_16, type, one_one_nat: nat).
% 2.58/1.93 thf(func_def_17, type, one_one_poly_nat: poly_nat).
% 2.58/1.93 thf(func_def_18, type, one_on501200385y_real: poly_poly_real).
% 2.58/1.93 thf(func_def_19, type, one_one_poly_real: poly_real).
% 2.58/1.93 thf(func_def_20, type, one_one_real: real).
% 2.58/1.93 thf(func_def_21, type, zero_zero_nat: nat).
% 2.58/1.93 thf(func_def_22, type, zero_zero_poly_nat: poly_nat).
% 2.58/1.93 thf(func_def_23, type, zero_z1059985641ly_nat: poly_poly_nat).
% 2.58/1.93 thf(func_def_24, type, zero_z935034829y_real: poly_poly_poly_real).
% 2.58/1.93 thf(func_def_25, type, zero_z1423781445y_real: poly_poly_real).
% 2.58/1.93 thf(func_def_26, type, zero_zero_poly_real: poly_real).
% 2.58/1.93 thf(func_def_27, type, zero_zero_real: real).
% 2.58/1.93 thf(func_def_28, type, ring_1897377867y_real: set_poly_poly_real).
% 2.58/1.93 thf(func_def_29, type, ring_1690226883y_real: set_poly_real).
% 2.58/1.93 thf(func_def_30, type, ring_1_Ints_real: set_real).
% 2.58/1.93 thf(func_def_31, type, suc: nat > nat).
% 2.58/1.93 thf(func_def_32, type, ord_less_nat: nat > nat > $o).
% 2.58/1.93 thf(func_def_33, type, ord_le38482960y_real: poly_poly_real > poly_poly_real > $o).
% 2.58/1.93 thf(func_def_34, type, ord_less_poly_real: poly_real > poly_real > $o).
% 2.58/1.93 thf(func_def_35, type, ord_less_real: real > real > $o).
% 2.58/1.93 thf(func_def_36, type, ord_less_eq_real: real > real > $o).
% 2.58/1.93 thf(func_def_37, type, algebraic_real: real > $o).
% 2.58/1.93 thf(func_def_38, type, algebraic_int_real: real > $o).
% 2.58/1.93 thf(func_def_39, type, cr_poly_real: (nat > real) > poly_real > $o).
% 2.58/1.93 thf(func_def_40, type, degree_nat: poly_nat > nat).
% 2.58/1.93 thf(func_def_41, type, degree_poly_nat: poly_poly_nat > nat).
% 2.58/1.93 thf(func_def_42, type, degree360860553y_real: poly_poly_poly_real > nat).
% 2.58/1.93 thf(func_def_43, type, degree_poly_real: poly_poly_real > nat).
% 2.58/1.93 thf(func_def_44, type, degree_real: poly_real > nat).
% 2.58/1.93 thf(func_def_45, type, divide924636027y_real: poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > nat > nat > poly_poly_poly_real).
% 2.58/1.93 thf(func_def_46, type, divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real).
% 2.58/1.93 thf(func_def_47, type, divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real).
% 2.58/1.93 thf(func_def_48, type, is_zero_nat: poly_nat > $o).
% 2.58/1.93 thf(func_def_49, type, is_zero_poly_real: poly_poly_real > $o).
% 2.58/1.93 thf(func_def_50, type, is_zero_real: poly_real > $o).
% 2.58/1.93 thf(func_def_51, type, order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat).
% 2.58/1.93 thf(func_def_52, type, order_poly_real: poly_real > poly_poly_real > nat).
% 2.58/1.93 thf(func_def_53, type, order_real: real > poly_real > nat).
% 2.58/1.93 thf(func_def_54, type, pcr_poly_nat_nat: (nat > nat > $o) > (nat > nat) > poly_nat > $o).
% 2.58/1.93 thf(func_def_55, type, pcr_po273983709ly_nat: (poly_nat > poly_nat > $o) > (nat > poly_nat) > poly_poly_nat > $o).
% 2.58/1.93 thf(func_def_56, type, pcr_po1200519205y_real: (poly_poly_real > poly_poly_real > $o) > (nat > poly_poly_real) > poly_poly_poly_real > $o).
% 2.58/1.93 thf(func_def_57, type, pcr_po1314690837y_real: (poly_real > poly_real > $o) > (nat > poly_real) > poly_poly_real > $o).
% 2.58/1.93 thf(func_def_58, type, pcr_poly_real_real: (real > real > $o) > (nat > real) > poly_real > $o).
% 2.58/1.93 thf(func_def_59, type, pderiv_nat: poly_nat > poly_nat).
% 2.58/1.93 thf(func_def_60, type, pderiv_poly_real: poly_poly_real > poly_poly_real).
% 2.58/1.93 thf(func_def_61, type, pderiv_real: poly_real > poly_real).
% 2.58/1.93 thf(func_def_62, type, poly_nat2: poly_nat > nat > nat).
% 2.58/1.93 thf(func_def_63, type, poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat).
% 2.58/1.93 thf(func_def_64, type, poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real).
% 2.58/1.93 thf(func_def_65, type, poly_poly_real2: poly_poly_real > poly_real > poly_real).
% 2.58/1.93 thf(func_def_66, type, poly_real2: poly_real > real > real).
% 2.58/1.93 thf(func_def_67, type, coeff_nat: poly_nat > nat > nat).
% 2.58/1.93 thf(func_def_68, type, coeff_poly_nat: poly_poly_nat > nat > poly_nat).
% 2.58/1.93 thf(func_def_69, type, coeff_poly_poly_real: poly_poly_poly_real > nat > poly_poly_real).
% 2.58/1.93 thf(func_def_70, type, coeff_poly_real: poly_poly_real > nat > poly_real).
% 2.58/1.93 thf(func_def_71, type, coeff_real: poly_real > nat > real).
% 2.58/1.93 thf(func_def_72, type, poly_cutoff_nat: nat > poly_nat > poly_nat).
% 2.58/1.93 thf(func_def_73, type, poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real).
% 2.58/1.93 thf(func_def_74, type, poly_cutoff_real: nat > poly_real > poly_real).
% 2.58/1.93 thf(func_def_75, type, poly_shift_nat: nat > poly_nat > poly_nat).
% 2.58/1.93 thf(func_def_76, type, poly_shift_poly_real: nat > poly_poly_real > poly_poly_real).
% 2.58/1.93 thf(func_def_77, type, poly_shift_real: nat > poly_real > poly_real).
% 2.58/1.93 thf(func_def_78, type, reflect_poly_nat: poly_nat > poly_nat).
% 2.58/1.93 thf(func_def_79, type, reflec781175074ly_nat: poly_poly_nat > poly_poly_nat).
% 2.58/1.93 thf(func_def_80, type, reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real).
% 2.58/1.93 thf(func_def_81, type, reflec1522834046y_real: poly_poly_real > poly_poly_real).
% 2.58/1.93 thf(func_def_82, type, reflect_poly_real: poly_real > poly_real).
% 2.58/1.93 thf(func_def_83, type, rsquar1555552848y_real: poly_poly_real > $o).
% 2.58/1.93 thf(func_def_84, type, rsquarefree_real: poly_real > $o).
% 2.58/1.93 thf(func_def_85, type, synthetic_div_nat: poly_nat > nat > poly_nat).
% 2.58/1.93 thf(func_def_86, type, synthe1498897281y_real: poly_poly_real > poly_real > poly_poly_real).
% 2.58/1.93 thf(func_def_87, type, synthetic_div_real: poly_real > real > poly_real).
% 2.58/1.93 thf(func_def_88, type, field_1537545994s_real: set_real).
% 2.58/1.93 thf(func_def_89, type, dvd_dvd_nat: nat > nat > $o).
% 2.58/1.93 thf(func_def_90, type, dvd_dvd_poly_nat: poly_nat > poly_nat > $o).
% 2.58/1.93 thf(func_def_91, type, dvd_dv1946063458y_real: poly_poly_real > poly_poly_real > $o).
% 2.58/1.93 thf(func_def_92, type, dvd_dvd_poly_real: poly_real > poly_real > $o).
% 2.58/1.93 thf(func_def_93, type, dvd_dvd_real: real > real > $o).
% 2.58/1.93 thf(func_def_94, type, collect_nat: (nat > $o) > set_nat).
% 2.58/1.93 thf(func_def_95, type, collec927113489y_real: (poly_poly_real > $o) > set_poly_poly_real).
% 2.58/1.93 thf(func_def_96, type, collect_poly_real: (poly_real > $o) > set_poly_real).
% 2.58/1.93 thf(func_def_97, type, collect_real: (real > $o) > set_real).
% 2.58/1.93 thf(func_def_98, type, member_nat: nat > set_nat > $o).
% 2.58/1.93 thf(func_def_99, type, member1159720147y_real: poly_poly_real > set_poly_poly_real > $o).
% 2.58/1.93 thf(func_def_100, type, member_poly_real: poly_real > set_poly_real > $o).
% 2.58/1.93 thf(func_def_101, type, member_real: real > set_real > $o).
% 2.58/1.93 thf(func_def_102, type, n: nat).
% 2.58/1.93 thf(func_def_103, type, p: poly_real).
% 2.58/1.93 thf(func_def_104, type, x: real).
% 2.58/1.93 thf(func_def_122, type, sK1: (nat > real) > nat).
% 2.58/1.93 thf(func_def_123, type, sK2: real > real).
% 2.58/1.93 thf(func_def_124, type, sK3: set_real > (nat > real > $o) > set_nat > real).
% 2.58/1.93 thf(func_def_125, type, sK4: set_real > (nat > real > $o) > set_nat > nat).
% 2.58/1.93 thf(func_def_126, type, sK5: poly_real).
% 2.58/1.93 thf(func_def_127, type, sK6: nat > nat).
% 2.58/1.93 thf(func_def_128, type, sK7: (real > poly_real > $o) > set_real > set_poly_real > real).
% 2.58/1.93 thf(func_def_129, type, sK8: (real > poly_real > $o) > set_real > set_poly_real > poly_real).
% 2.58/1.93 thf(func_def_130, type, sK9: (nat > $o) > nat).
% 2.58/1.93 thf(func_def_131, type, sK10: (poly_real > $o) > poly_real).
% 2.58/1.93 thf(func_def_132, type, sK11: set_real > (poly_real > real > $o) > set_poly_real > real).
% 2.58/1.93 thf(func_def_133, type, sK12: set_real > (poly_real > real > $o) > set_poly_real > poly_real).
% 2.58/1.93 thf(func_def_134, type, sK13: nat > nat > nat).
% 2.58/1.93 thf(func_def_135, type, sK14: nat > nat > nat).
% 2.58/1.93 thf(func_def_136, type, sK15: real > real > real).
% 2.58/1.93 thf(func_def_137, type, sK16: poly_real > nat).
% 2.58/1.93 thf(func_def_138, type, sK17: poly_poly_real > nat > nat).
% 2.58/1.93 thf(func_def_139, type, sK18: set_poly_real > (poly_real > poly_real > $o) > set_poly_real > poly_real).
% 2.58/1.93 thf(func_def_140, type, sK19: real > real).
% 2.58/1.93 thf(func_def_141, type, sK20: (nat > $o) > nat > nat).
% 2.58/1.93 thf(func_def_142, type, sK21: (nat > $o) > nat > nat).
% 2.58/1.93 thf(func_def_143, type, sK22: poly_real).
% 2.58/1.93 thf(func_def_144, type, sK23: (nat > $o) > nat).
% 2.58/1.93 thf(func_def_145, type, sK24: poly_real > poly_real > nat).
% 2.58/1.93 thf(func_def_146, type, sK25: set_nat > (nat > poly_real > $o) > set_poly_real > nat).
% 2.58/1.93 thf(func_def_147, type, sK26: (nat > $o) > nat).
% 2.58/1.93 thf(func_def_148, type, sK27: (real > nat > $o) > set_nat > set_real > real).
% 2.58/1.93 thf(func_def_149, type, sK28: (real > nat > $o) > set_nat > set_real > nat).
% 2.58/1.93 thf(func_def_150, type, sK29: poly_real > real).
% 2.58/1.93 thf(func_def_151, type, sK30: real > poly_real > real > real).
% 2.58/1.93 thf(func_def_152, type, sK31: (nat > $o) > (nat > $o) > nat).
% 2.58/1.93 thf(func_def_153, type, sK32: nat > (nat > $o) > nat).
% 2.58/1.93 thf(func_def_154, type, sK33: (nat > $o) > nat).
% 2.58/1.93 thf(func_def_155, type, sK34: (poly_real > $o) > (poly_real > $o) > poly_real).
% 2.58/1.93 thf(func_def_156, type, sK35: nat > (nat > $o) > nat).
% 2.58/1.93 thf(func_def_157, type, sK36: nat > (nat > $o) > nat).
% 2.58/1.93 thf(func_def_158, type, sK37: (nat > nat > $o) > nat).
% 2.58/1.93 thf(func_def_159, type, sK38: (nat > nat > $o) > nat).
% 2.58/1.93 thf(func_def_160, type, sK39: (nat > nat > $o) > nat).
% 2.58/1.93 thf(func_def_161, type, sK40: (nat > nat > $o) > nat).
% 2.58/1.93 thf(func_def_162, type, sK41: real > poly_real).
% 2.58/1.93 thf(func_def_163, type, sK42: (nat > $o) > nat).
% 2.58/1.93 thf(func_def_164, type, sK43: poly_poly_real > poly_real).
% 2.58/1.93 thf(func_def_165, type, sK44: (nat > $o) > nat > nat).
% 2.58/1.93 thf(func_def_166, type, sK45: (nat > $o) > nat > nat).
% 2.58/1.93 thf(func_def_167, type, sK46: (nat > nat > $o) > nat).
% 2.58/1.93 thf(func_def_168, type, sK47: (nat > nat > $o) > nat).
% 2.58/1.93 thf(func_def_169, type, sK48: (nat > nat > $o) > nat).
% 2.58/1.93 thf(func_def_170, type, sK49: (nat > nat > $o) > nat).
% 2.58/1.93 thf(func_def_171, type, sK50: (real > $o) > real).
% 2.58/1.93 thf(func_def_172, type, sK51: (real > $o) > poly_real).
% 2.58/1.93 thf(func_def_173, type, sK52: (real > $o) > (real > $o) > real).
% 2.58/1.93 thf(func_def_174, type, sK53: (nat > $o) > nat).
% 2.58/1.93 thf(func_def_175, type, sK54: nat > (nat > $o) > nat).
% 2.58/1.93 thf(func_def_176, type, sK55: real > poly_real).
% 2.58/1.93 thf(func_def_177, type, sK56: poly_real > nat).
% 2.58/1.93 thf(func_def_178, type, sK57: nat > nat > nat).
% 2.58/1.93 thf(func_def_179, type, sK58: (nat > $o) > nat).
% 2.58/1.93 thf(func_def_180, type, sK59: (real > $o) > real).
% 2.58/1.93 thf(func_def_181, type, sK60: set_nat > (poly_real > nat > $o) > set_poly_real > poly_real).
% 2.58/1.93 thf(func_def_182, type, sK61: nat > nat).
% 2.58/1.93 thf(func_def_183, type, sK62: (nat > nat) > nat).
% 2.58/1.93 thf(func_def_184, type, sK63: set_nat > (nat > nat > $o) > set_nat > nat).
% 2.58/1.93 thf(func_def_185, type, sK64: set_real > set_real > (real > real > $o) > real).
% 2.58/1.93 thf(func_def_186, type, ph65: !>[X0: $tType]:(X0)).
% 2.58/1.93 thf(func_def_187, type, sK66: set_real > real).
% 2.58/1.93 thf(func_def_188, type, sK67: real).
% 2.58/1.93 thf(func_def_189, type, sK68: poly_real > real).
% 2.58/1.93 thf(f3552,plain,(
% 2.58/1.93 $false),
% 2.58/1.93 inference(avatar_sat_refutation,[status(thm)],[f3517,f3531,f3551])).
% 2.58/1.93 thf(f3551,plain,(
% 2.58/1.93 ~spl0_9),
% 2.58/1.93 inference(avatar_contradiction_clause,[status(thm)],[f3550])).
% 2.58/1.93 thf(f3550,plain,(
% 2.58/1.93 $false | ~spl0_9),
% 2.58/1.93 inference(trivial_inequality_removal,[status(thm)],[f3549])).
% 2.58/1.93 thf(f3549,plain,(
% 2.58/1.93 ((pderiv_real @ one_one_poly_real) != (pderiv_real @ one_one_poly_real)) | ~spl0_9),
% 2.58/1.93 inference(superposition,[status(thm)],[f1272,f3514])).
% 2.58/1.93 thf(f3514,plain,(
% 2.58/1.93 ((pderiv_real @ p) = (pderiv_real @ one_one_poly_real)) | ~spl0_9),
% 2.58/1.93 inference(avatar_component_clause,[status(thm)],[f3512])).
% 2.58/1.93 thf(f3512,definition,(
% 2.58/1.93 spl0_9 <=> ((pderiv_real @ p) = (pderiv_real @ one_one_poly_real))),
% 2.58/1.93 introduced(definition,[new_symbols(naming,[spl0_9])],[avatar_definition])).
% 2.58/1.93 thf(f1272,plain,(
% 2.58/1.93 ((pderiv_real @ p) != (pderiv_real @ one_one_poly_real))),
% 2.58/1.93 inference(definition_unfolding,[status(thm)],[f1145,f1081])).
% 2.58/1.93 thf(f1081,plain,(
% 2.58/1.93 (zero_zero_poly_real = (pderiv_real @ one_one_poly_real))),
% 2.58/1.93 inference(cnf_transformation,[status(thm)],[f855])).
% 2.58/1.93 thf(f855,plain,(
% 2.58/1.93 (zero_zero_poly_real = (pderiv_real @ one_one_poly_real))),
% 2.58/1.93 inference(fool_elimination,[status(thm)],[f172])).
% 2.58/1.93 thf(f172,axiom,(
% 2.58/1.93 (zero_zero_poly_real = (pderiv_real @ one_one_poly_real))),
% 2.58/1.93 file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',fact_171_pderiv__1)).
% 2.58/1.93 thf(f1145,plain,(
% 2.58/1.93 (zero_zero_poly_real != (pderiv_real @ p))),
% 2.58/1.93 inference(cnf_transformation,[status(thm)],[f896])).
% 2.58/1.93 thf(f896,plain,(
% 2.58/1.93 (zero_zero_poly_real != (pderiv_real @ p))),
% 2.58/1.93 inference(fool_elimination,[status(thm)],[f5])).
% 2.58/1.93 thf(f5,axiom,(
% 2.58/1.93 (zero_zero_poly_real != (pderiv_real @ p))),
% 2.58/1.93 file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',fact_4__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062)).
% 2.58/1.93 thf(f3531,plain,(
% 2.58/1.93 spl0_9 | ~spl0_8),
% 2.58/1.93 inference(avatar_split_clause,[status(thm)],[f3530,f3508,f3512])).
% 2.58/1.93 thf(f3508,definition,(
% 2.58/1.93 spl0_8 <=> ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ (pderiv_real @ p))))),
% 2.58/1.93 introduced(definition,[new_symbols(naming,[spl0_8])],[avatar_definition])).
% 2.58/1.93 thf(f3530,plain,(
% 2.58/1.93 ((pderiv_real @ p) = (pderiv_real @ one_one_poly_real)) | ~spl0_8),
% 2.58/1.93 inference(trivial_inequality_removal,[status(thm)],[f3529])).
% 2.58/1.93 thf(f3529,plain,(
% 2.58/1.93 ((poly_real2 @ p @ x) != (poly_real2 @ p @ x)) | ((pderiv_real @ p) = (pderiv_real @ one_one_poly_real)) | ~spl0_8),
% 2.58/1.93 inference(duplicate_literal_removal,[status(thm)],[f3528])).
% 2.58/1.93 thf(f3528,plain,(
% 2.58/1.93 ((pderiv_real @ p) = (pderiv_real @ one_one_poly_real)) | ((poly_real2 @ p @ x) != (poly_real2 @ p @ x)) | ((poly_real2 @ p @ x) != (poly_real2 @ p @ x)) | ~spl0_8),
% 2.58/1.93 inference(superposition,[status(thm)],[f3490,f3510])).
% 2.58/1.93 thf(f3510,plain,(
% 2.58/1.93 ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ (pderiv_real @ p)))) | ~spl0_8),
% 2.58/1.93 inference(avatar_component_clause,[status(thm)],[f3508])).
% 2.58/1.93 thf(f3490,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (((poly_real2 @ p @ x) != (poly_real2 @ (pderiv_real @ p) @ (sK68 @ X0))) | ((pderiv_real @ one_one_poly_real) = X0) | ((poly_real2 @ p @ x) != (poly_real2 @ X0 @ (sK68 @ X0)))) )),
% 2.58/1.93 inference(equality_proxy_clausification,[status(thm)],[f3489])).
% 2.58/1.93 thf(f3489,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (((pderiv_real @ one_one_poly_real) = X0) | ($false = ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ X0)))) | ((poly_real2 @ p @ x) != (poly_real2 @ X0 @ (sK68 @ X0)))) )),
% 2.58/1.93 inference(equality_proxy_clausification,[status(thm)],[f3488])).
% 2.58/1.93 thf(f3488,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (((pderiv_real @ one_one_poly_real) = X0) | (((poly_real2 @ X0 @ (sK68 @ X0)) = (poly_real2 @ p @ x)) = $false) | ($false = ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ X0))))) )),
% 2.58/1.93 inference(binary_proxy_clausification,[status(thm)],[f3482])).
% 2.58/1.93 thf(f3482,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (((pderiv_real @ one_one_poly_real) = X0) | (((poly_real2 @ X0 @ (sK68 @ X0)) = (poly_real2 @ p @ x)) != ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ X0))))) )),
% 2.58/1.93 inference(beta_eta_normalization,[status(thm)],[f3480])).
% 2.58/1.93 thf(f3480,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (((pderiv_real @ one_one_poly_real) = X0) | (((^[Y0 : real]: ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ Y0))) @ (sK68 @ X0)) != ((^[Y0 : real]: ((poly_real2 @ X0 @ Y0) = (poly_real2 @ p @ x))) @ (sK68 @ X0)))) )),
% 2.58/1.93 inference(negative_extensionality,[status(thm)],[f3371])).
% 2.58/1.93 thf(f3371,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (((^[Y0 : real]: ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ Y0))) != (^[Y0 : real]: ((poly_real2 @ X0 @ Y0) = (poly_real2 @ p @ x)))) | ((pderiv_real @ one_one_poly_real) = X0)) )),
% 2.58/1.93 inference(trivial_inequality_removal,[status(thm)],[f3369])).
% 2.58/1.93 thf(f3369,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (($false = $true) | ((^[Y0 : real]: ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ Y0))) != (^[Y0 : real]: ((poly_real2 @ X0 @ Y0) = (poly_real2 @ p @ x)))) | ((pderiv_real @ one_one_poly_real) = X0)) )),
% 2.58/1.93 inference(constrained_superposition,[status(thm)],[f2088,f1413])).
% 2.58/1.93 thf(f1413,plain,(
% 2.58/1.93 ((finite_finite_real @ (collect_real @ (^[Y0 : real]: ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ Y0))))) = $false)),
% 2.58/1.93 inference(not_proxy_clausification,[status(thm)],[f1222])).
% 2.58/1.93 thf(f1222,plain,(
% 2.58/1.93 ((~ (finite_finite_real @ (collect_real @ (^[Y0 : real]: ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ Y0)))))) = $true)),
% 2.58/1.93 inference(definition_unfolding,[status(thm)],[f1034,f1078])).
% 2.58/1.93 thf(f1078,plain,(
% 2.58/1.93 (zero_zero_real = (poly_real2 @ p @ x))),
% 2.58/1.93 inference(cnf_transformation,[status(thm)],[f409])).
% 2.58/1.93 thf(f409,plain,(
% 2.58/1.93 (zero_zero_real = (poly_real2 @ p @ x))),
% 2.58/1.93 inference(fool_elimination,[status(thm)],[f6])).
% 2.58/1.93 thf(f6,axiom,(
% 2.58/1.93 (zero_zero_real = (poly_real2 @ p @ x))),
% 2.58/1.93 file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',fact_5_p_I3_J)).
% 2.58/1.93 thf(f1034,plain,(
% 2.58/1.93 ((~ (finite_finite_real @ (collect_real @ (^[Y0 : real]: (zero_zero_real = (poly_real2 @ (pderiv_real @ p) @ Y0)))))) = $true)),
% 2.58/1.93 inference(cnf_transformation,[status(thm)],[f816])).
% 2.58/1.93 thf(f816,plain,(
% 2.58/1.93 ((~ (finite_finite_real @ (collect_real @ (^[Y0 : real]: (zero_zero_real = (poly_real2 @ (pderiv_real @ p) @ Y0)))))) = $true)),
% 2.58/1.93 inference(fool_elimination,[status(thm)],[f815])).
% 2.58/1.93 thf(f815,plain,(
% 2.58/1.93 ~(finite_finite_real @ (collect_real @ (^[X0 : real] : ((zero_zero_real = (poly_real2 @ (pderiv_real @ p) @ X0))))))),
% 2.58/1.93 inference(rectify,[status(thm)],[f346])).
% 2.58/1.93 thf(f346,negated_conjecture,(
% 2.58/1.93 ~(finite_finite_real @ (collect_real @ (^[X1 : real] : ((zero_zero_real = (poly_real2 @ (pderiv_real @ p) @ X1))))))),
% 2.58/1.93 inference(negated_conjecture,[status(cth)],[f345])).
% 2.58/1.93 thf(f345,conjecture,(
% 2.58/1.93 (finite_finite_real @ (collect_real @ (^[X1 : real] : ((zero_zero_real = (poly_real2 @ (pderiv_real @ p) @ X1))))))),
% 2.58/1.93 file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',conj_0)).
% 2.58/1.93 thf(f2088,plain,(
% 2.58/1.93 ( ! [X1 : poly_real] : (((finite_finite_real @ (collect_real @ (^[Y0 : real]: ((poly_real2 @ X1 @ Y0) = (poly_real2 @ p @ x))))) = $true) | ((pderiv_real @ one_one_poly_real) = X1)) )),
% 2.58/1.93 inference(equality_proxy_clausification,[status(thm)],[f2087])).
% 2.58/1.93 thf(f2087,plain,(
% 2.58/1.93 ( ! [X1 : poly_real] : (((finite_finite_real @ (collect_real @ (^[Y0 : real]: ((poly_real2 @ X1 @ Y0) = (poly_real2 @ p @ x))))) = $true) | (((pderiv_real @ one_one_poly_real) = X1) = $true)) )),
% 2.58/1.93 inference(not_proxy_clausification,[status(thm)],[f2086])).
% 2.58/1.93 thf(f2086,plain,(
% 2.58/1.93 ( ! [X1 : poly_real] : (((finite_finite_real @ (collect_real @ (^[Y0 : real]: ((poly_real2 @ X1 @ Y0) = (poly_real2 @ p @ x))))) = $true) | ((~ ((pderiv_real @ one_one_poly_real) = X1)) = $false)) )),
% 2.58/1.93 inference(binary_proxy_clausification,[status(thm)],[f2085])).
% 2.58/1.93 thf(f2085,plain,(
% 2.58/1.93 ( ! [X1 : poly_real] : (($true = ((~ ((pderiv_real @ one_one_poly_real) = X1)) => (finite_finite_real @ (collect_real @ (^[Y0 : real]: ((poly_real2 @ X1 @ Y0) = (poly_real2 @ p @ x)))))))) )),
% 2.58/1.93 inference(beta_eta_normalization,[status(thm)],[f2084])).
% 2.58/1.93 thf(f2084,plain,(
% 2.58/1.93 ( ! [X1 : poly_real] : ((((^[Y0 : poly_real]: ((~ ((pderiv_real @ one_one_poly_real) = Y0)) => (finite_finite_real @ (collect_real @ (^[Y1 : real]: ((poly_real2 @ Y0 @ Y1) = (poly_real2 @ p @ x))))))) @ X1) = $true)) )),
% 2.58/1.93 inference(pi_clausification,[status(thm)],[f1235])).
% 2.58/1.93 thf(f1235,plain,(
% 2.58/1.93 ((!! @ poly_real @ (^[Y0 : poly_real]: ((~ ((pderiv_real @ one_one_poly_real) = Y0)) => (finite_finite_real @ (collect_real @ (^[Y1 : real]: ((poly_real2 @ Y0 @ Y1) = (poly_real2 @ p @ x)))))))) = $true)),
% 2.58/1.93 inference(definition_unfolding,[status(thm)],[f1061,f1081,f1078])).
% 2.58/1.93 thf(f1061,plain,(
% 2.58/1.93 ((!! @ poly_real @ (^[Y0 : poly_real]: ((~ (zero_zero_poly_real = Y0)) => (finite_finite_real @ (collect_real @ (^[Y1 : real]: ((poly_real2 @ Y0 @ Y1) = zero_zero_real))))))) = $true)),
% 2.58/1.93 inference(cnf_transformation,[status(thm)],[f614])).
% 2.58/1.93 thf(f614,plain,(
% 2.58/1.93 ((!! @ poly_real @ (^[Y0 : poly_real]: ((~ (zero_zero_poly_real = Y0)) => (finite_finite_real @ (collect_real @ (^[Y1 : real]: ((poly_real2 @ Y0 @ Y1) = zero_zero_real))))))) = $true)),
% 2.58/1.93 inference(fool_elimination,[status(thm)],[f613])).
% 2.58/1.93 thf(f613,plain,(
% 2.58/1.93 ! [X0 : poly_real] : ((zero_zero_poly_real != X0) => (finite_finite_real @ (collect_real @ (^[X1 : real] : (((poly_real2 @ X0 @ X1) = zero_zero_real))))))),
% 2.58/1.93 inference(rectify,[status(thm)],[f3])).
% 2.58/1.93 thf(f3,axiom,(
% 2.58/1.93 ! [X0 : poly_real] : ((zero_zero_poly_real != X0) => (finite_finite_real @ (collect_real @ (^[X1 : real] : (((poly_real2 @ X0 @ X1) = zero_zero_real))))))),
% 2.58/1.93 file('/export/starexec/sandbox2/tmp/tmp.Sdgc6GdWVc/DTF2THF_11513.p',fact_2_poly__roots__finite)).
% 2.58/1.93 thf(f3517,plain,(
% 2.58/1.93 spl0_9 | spl0_8),
% 2.58/1.93 inference(avatar_split_clause,[status(thm)],[f3502,f3508,f3512])).
% 2.58/1.93 thf(f3502,plain,(
% 2.58/1.93 ((pderiv_real @ p) = (pderiv_real @ one_one_poly_real)) | ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ (pderiv_real @ p))))),
% 2.58/1.93 inference(trivial_inequality_removal,[status(thm)],[f3501])).
% 2.58/1.93 thf(f3501,plain,(
% 2.58/1.93 ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ (pderiv_real @ p)))) | ((poly_real2 @ p @ x) != (poly_real2 @ p @ x)) | ((pderiv_real @ p) = (pderiv_real @ one_one_poly_real))),
% 2.58/1.93 inference(equality_factoring,[status(thm)],[f3492])).
% 2.58/1.93 thf(f3492,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ X0))) | ((poly_real2 @ p @ x) = (poly_real2 @ X0 @ (sK68 @ X0))) | ((pderiv_real @ one_one_poly_real) = X0)) )),
% 2.58/1.93 inference(equality_proxy_clausification,[status(thm)],[f3491])).
% 2.58/1.93 thf(f3491,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (((pderiv_real @ one_one_poly_real) = X0) | ((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ X0))) | (((poly_real2 @ X0 @ (sK68 @ X0)) = (poly_real2 @ p @ x)) = $true)) )),
% 2.58/1.93 inference(equality_proxy_clausification,[status(thm)],[f3487])).
% 2.58/1.93 thf(f3487,plain,(
% 2.58/1.93 ( ! [X0 : poly_real] : (((pderiv_real @ one_one_poly_real) = X0) | (((poly_real2 @ p @ x) = (poly_real2 @ (pderiv_real @ p) @ (sK68 @ X0))) = $true) | (((poly_real2 @ X0 @ (sK68 @ X0)) = (poly_real2 @ p @ x)) = $true)) )),
% 2.58/1.93 inference(binary_proxy_clausification,[status(thm)],[f3482])).
% 2.58/1.93 % SZS output end Proof for DTF2THF_11513
% 2.58/1.93 % (11881)------------------------------
% 2.58/1.93 % (11881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.58/1.93 % (11881)Termination reason: Refutation
% 2.58/1.93
% 2.58/1.93 % (11881)Memory used [KB]: 7803
% 2.58/1.93 % (11881)Time elapsed: 0.109 s
% 2.58/1.93 % (11881)Instructions burned: 373 (million)
% 2.58/1.93 % (11881)------------------------------
% 2.58/1.93 % (11881)------------------------------
% 2.58/1.93 % (11740)Success in time 0.397 s
%------------------------------------------------------------------------------