%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU809^2 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n012.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 09:37:23 AM UTC 2026
% Result : Theorem 1.42s 1.27s
% Output : Refutation 1.42s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 2
% Syntax : Number of formulae : 13 ( 10 unt; 0 typ; 0 def)
% Number of atoms : 76 ( 9 equ; 0 cnn)
% Maximal formula atoms : 5 ( 5 avg)
% Number of connectives : 107 ( 6 ~; 0 |; 0 &; 78 @)
% ( 0 <=>; 17 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 8 ( 4 usr; 5 con; 0-2 aty)
% ( 6 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 11 ( 6 ^; 5 !; 0 ?; 11 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
setadjoin: $i > $i > $i ).
thf(f14,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f13]) ).
thf(f13,plain,
$true = $false,
inference(boolean_simplification,[status(thm)],[f12]) ).
thf(f12,plain,
( ( ~ $true ) = $true ),
inference(boolean_simplification,[status(thm)],[f11]) ).
thf(f11,plain,
( $true
= ( ~ ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ omega )
=> ( in @ ( setadjoin @ Y0 @ Y0 ) @ omega ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ omega )
=> ( in @ ( setadjoin @ Y0 @ Y0 ) @ omega ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f9,f10]) ).
thf(f10,plain,
( omegaSAx
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ omega )
=> ( in @ ( setadjoin @ Y0 @ Y0 ) @ omega ) ) ) ),
inference(cnf_transformation,[status(thm)],[f6]) ).
thf(f6,plain,
( omegaSAx
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ omega )
=> ( in @ ( setadjoin @ Y0 @ Y0 ) @ omega ) ) ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
! [X0: $i] :
( ( ( in @ X0 @ omega )
=> ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) )
= omegaSAx ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: $i] :
( ( ( in @ X0 @ omega )
=> ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) )
= omegaSAx ),
file('/export/starexec/sandbox/tmp/tmp.0kYmzGpyRn/DTF2THF_24169.p',omegaSAx) ).
thf(f9,plain,
( ( ~ ( omegaSAx
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ omega )
=> ( in @ ( setadjoin @ Y0 @ Y0 ) @ omega ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( ( ~ ( omegaSAx
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ omega )
=> ( in @ ( setadjoin @ Y0 @ Y0 ) @ omega ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
~ ( omegaSAx
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
~ ( omegaSAx
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
( omegaSAx
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ),
file('/export/starexec/sandbox/tmp/tmp.0kYmzGpyRn/DTF2THF_24169.p',omegaSclos) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : SEU809^2 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.07 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.25 % Computer : n012.cluster.edu
% 0.07/0.25 % Model : x86_64 x86_64
% 0.07/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.25 % Memory : 8042.1875MB
% 0.07/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.25 % CPULimit : 300
% 0.07/0.25 % WCLimit : 300
% 0.07/0.25 % DateTime : Tue Feb 24 12:34:30 EST 2026
% 0.07/0.25 % CPUTime :
% 0.07/0.25 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 ---- Original DTF file ---
% 0.18/0.34 thf(spec,logic,$$dhol).
% 0.18/0.34 %------------------------------------------------------------------------------
% 0.18/0.34 % File : SEU809^2 : TPTP v9.2.1. Released v3.7.0.
% 0.18/0.34 % Domain : Set Theory
% 0.18/0.34 % Problem : Omega and Peano
% 0.18/0.34 % Version : Especial > Reduced > Especial.
% 0.18/0.34 % English : (! x:i.in x omega -> in (setadjoin x x) omega)
% 0.18/0.34
% 0.18/0.34 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.18/0.34 % Source : [Bro08]
% 0.18/0.34 % Names : ZFC311l [Bro08]
% 0.18/0.34
% 0.18/0.34 % Status : Theorem
% 0.18/0.34 % Rating : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v6.0.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.00 v3.7.0
% 0.18/0.34 % Syntax : Number of formulae : 6 ( 1 unt; 4 typ; 1 def)
% 0.18/0.34 % Number of atoms : 7 ( 1 equ; 0 cnn)
% 0.18/0.34 % Maximal formula atoms : 3 ( 3 avg)
% 0.18/0.34 % Number of connectives : 15 ( 0 ~; 0 |; 0 &; 12 @)
% 0.18/0.34 % ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% 0.18/0.34 % Maximal formula depth : 8 ( 5 avg)
% 0.18/0.34 % Number of types : 2 ( 0 usr)
% 0.18/0.34 % Number of type conns : 4 ( 4 >; 0 *; 0 +; 0 <<)
% 0.18/0.34 % Number of symbols : 5 ( 4 usr; 2 con; 0-2 aty)
% 0.18/0.34 % Number of variables : 2 ( 0 ^; 2 !; 0 ?; 2 :)
% 0.18/0.34 % SPC : TH0_THM_EQU_NAR
% 0.18/0.34
% 0.18/0.34 % Comments : http://mathgate.info/detsetitem.php?id=442
% 0.18/0.34 %------------------------------------------------------------------------------
% 0.18/0.34 thf(in_type,type,
% 0.18/0.34 in: $i > $i > $o ).
% 0.18/0.34
% 0.18/0.34 thf(setadjoin_type,type,
% 0.18/0.34 setadjoin: $i > $i > $i ).
% 0.18/0.34
% 0.18/0.34 thf(omega_type,type,
% 0.18/0.34 omega: $i ).
% 0.18/0.34
% 0.18/0.34 thf(omegaSAx_type,type,
% 0.18/0.34 omegaSAx: $o ).
% 0.18/0.34
% 0.18/0.34 thf(omegaSAx,definition,
% 0.18/0.34 ( omegaSAx
% 0.18/0.34 = ( ! [Xx: $i] :
% 0.18/0.34 ( ( in @ Xx @ omega )
% 0.18/0.34 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.18/0.34
% 0.18/0.34 thf(omegaSclos,conjecture,
% 0.18/0.34 ( omegaSAx
% 0.18/0.34 => ! [Xx: $i] :
% 0.18/0.34 ( ( in @ Xx @ omega )
% 0.18/0.34 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ).
% 0.18/0.34
% 0.18/0.34 %------------------------------------------------------------------------------
% 0.18/0.34 ------------------------
% 1.42/1.24 ---- Embedded in THF ---
% 1.42/1.24 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.42/1.24 %%% Generated on Tue Feb 24 12:34:31 EST 2026
% 1.42/1.24 %%% using '$$dhol' embedding, version 1.3.0.
% 1.42/1.24 %%% Logic specification used:
% 1.42/1.24 %%% thf(spec, logic, $$dhol).
% 1.42/1.24
% 1.42/1.24 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.0kYmzGpyRn/DTF2DTF24169.p
% See solution above
% 1.42/1.24 ------------------------
% 1.42/1.25 ---- Cleaned THF ---
% 1.42/1.25 thf(in_type,type,
% 1.42/1.25 in: $i > $i > $o ).
% 1.42/1.25
% 1.42/1.25 thf(setadjoin_type,type,
% 1.42/1.25 setadjoin: $i > $i > $i ).
% 1.42/1.25
% 1.42/1.25 thf(omega_type,type,
% 1.42/1.25 omega: $i ).
% 1.42/1.25
% 1.42/1.25 thf(omegaSAx_type,type,
% 1.42/1.25 omegaSAx: $o ).
% 1.42/1.25
% 1.42/1.25 thf(omegaSAx,definition,
% 1.42/1.25 ( omegaSAx
% 1.42/1.25 = ( ! [Xx: $i] :
% 1.42/1.25 ( ( in @ Xx @ omega )
% 1.42/1.25 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.42/1.25
% 1.42/1.25 thf(omegaSclos,conjecture,
% 1.42/1.25 ( omegaSAx
% 1.42/1.25 => ! [Xx: $i] :
% 1.42/1.25 ( ( in @ Xx @ omega )
% 1.42/1.25 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ).
% 1.42/1.25 ------------------------
% 1.42/1.27 % (24268)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_24169 for (3000ds/27Mi)
% 1.42/1.27 % (24270)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_24169 for (3000ds/2Mi)
% 1.42/1.27 % (24266)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24169 for (3000ds/183Mi)
% 1.42/1.27 % (24267)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_24169 for (3000ds/4Mi)
% 1.42/1.27 % (24269)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24169 for (3000ds/2Mi)
% 1.42/1.27 % (24272)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24169 for (3000ds/18Mi)
% 1.42/1.27 % (24271)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_24169 for (3000ds/275Mi)
% 1.42/1.27 % (24270)First to succeed.
% 1.42/1.27 % (24269)Instruction limit reached!
% 1.42/1.27 % (24269)------------------------------
% 1.42/1.27 % (24269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.42/1.27 % (24269)Termination reason: Unknown
% 1.42/1.27 % (24269)Termination phase: Saturation
% 1.42/1.27
% 1.42/1.27 % (24269)Memory used [KB]: 5500
% 1.42/1.27 % (24269)Time elapsed: 0.003 s
% 1.42/1.27 % (24269)Instructions burned: 2 (million)
% 1.42/1.27 % (24269)------------------------------
% 1.42/1.27 % (24269)------------------------------
% 1.42/1.27 % (24271)Also succeeded, but the first one will report.
% 1.42/1.27 % (24270)Refutation found. Thanks to Tanya!
% 1.42/1.27 % SZS status Theorem for DTF2THF_24169
% 1.42/1.27 % SZS output start Proof for DTF2THF_24169
% 1.42/1.27 thf(func_def_0, type, in: $i > $i > $o).
% 1.42/1.27 thf(func_def_1, type, setadjoin: $i > $i > $i).
% 1.42/1.27 thf(f14,plain,(
% 1.42/1.27 $false),
% 1.42/1.27 inference(trivial_inequality_removal,[status(thm)],[f13])).
% 1.42/1.27 thf(f13,plain,(
% 1.42/1.27 ($true = $false)),
% 1.42/1.27 inference(boolean_simplification,[status(thm)],[f12])).
% 1.42/1.27 thf(f12,plain,(
% 1.42/1.27 ((~ $true) = $true)),
% 1.42/1.27 inference(boolean_simplification,[status(thm)],[f11])).
% 1.42/1.27 thf(f11,plain,(
% 1.42/1.27 ($true = (~ ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ omega) => (in @ (setadjoin @ Y0 @ Y0) @ omega)))) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ omega) => (in @ (setadjoin @ Y0 @ Y0) @ omega)))))))),
% 1.42/1.27 inference(definition_unfolding,[status(thm)],[f9,f10])).
% 1.42/1.27 thf(f10,plain,(
% 1.42/1.27 (omegaSAx = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ omega) => (in @ (setadjoin @ Y0 @ Y0) @ omega)))))),
% 1.42/1.27 inference(cnf_transformation,[status(thm)],[f6])).
% 1.42/1.27 thf(f6,plain,(
% 1.42/1.27 (omegaSAx = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ omega) => (in @ (setadjoin @ Y0 @ Y0) @ omega)))))),
% 1.42/1.27 inference(fool_elimination,[status(thm)],[f5])).
% 1.42/1.27 thf(f5,plain,(
% 1.42/1.27 (! [X0 : $i] : ((in @ X0 @ omega) => (in @ (setadjoin @ X0 @ X0) @ omega)) = omegaSAx)),
% 1.42/1.27 inference(rectify,[status(thm)],[f1])).
% 1.42/1.27 thf(f1,axiom,(
% 1.42/1.27 (! [X0 : $i] : ((in @ X0 @ omega) => (in @ (setadjoin @ X0 @ X0) @ omega)) = omegaSAx)),
% 1.42/1.27 file('/export/starexec/sandbox/tmp/tmp.0kYmzGpyRn/DTF2THF_24169.p',omegaSAx)).
% 1.42/1.27 thf(f9,plain,(
% 1.42/1.27 ((~ (omegaSAx => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ omega) => (in @ (setadjoin @ Y0 @ Y0) @ omega)))))) = $true)),
% 1.42/1.27 inference(cnf_transformation,[status(thm)],[f8])).
% 1.42/1.27 thf(f8,plain,(
% 1.42/1.27 ((~ (omegaSAx => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ omega) => (in @ (setadjoin @ Y0 @ Y0) @ omega)))))) = $true)),
% 1.42/1.27 inference(fool_elimination,[status(thm)],[f7])).
% 1.42/1.27 thf(f7,plain,(
% 1.42/1.27 ~(omegaSAx => ! [X0 : $i] : ((in @ X0 @ omega) => (in @ (setadjoin @ X0 @ X0) @ omega)))),
% 1.42/1.27 inference(rectify,[status(thm)],[f3])).
% 1.42/1.27 thf(f3,negated_conjecture,(
% 1.42/1.27 ~(omegaSAx => ! [X0 : $i] : ((in @ X0 @ omega) => (in @ (setadjoin @ X0 @ X0) @ omega)))),
% 1.42/1.27 inference(negated_conjecture,[status(cth)],[f2])).
% 1.42/1.27 thf(f2,conjecture,(
% 1.42/1.27 omegaSAx => ! [X0 : $i] : ((in @ X0 @ omega) => (in @ (setadjoin @ X0 @ X0) @ omega))),
% 1.42/1.27 file('/export/starexec/sandbox/tmp/tmp.0kYmzGpyRn/DTF2THF_24169.p',omegaSclos)).
% 1.42/1.27 % SZS output end Proof for DTF2THF_24169
% 1.42/1.27 % (24270)------------------------------
% 1.42/1.27 % (24270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.42/1.27 % (24270)Termination reason: Refutation
% 1.42/1.27
% 1.42/1.27 % (24270)Memory used [KB]: 5373
% 1.42/1.27 % (24270)Time elapsed: 0.003 s
% 1.42/1.27 % (24270)Instructions burned: 1 (million)
% 1.42/1.27 % (24270)------------------------------
% 1.42/1.27 % (24270)------------------------------
% 1.42/1.27 % (24265)Success in time 0.017 s
%------------------------------------------------------------------------------