%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : NUN023^2 : TPTP v9.3.1. Released v6.4.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:19:48 AM UTC 2026
% Result : Theorem 5.20s 2.71s
% Output : Refutation 5.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 1
% Syntax : Number of formulae : 39 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 152 ( 125 equ; 0 cnn)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 393 ( 51 ~; 29 |; 32 &; 260 @)
% ( 0 <=>; 21 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 18 ( 18 >; 0 *; 0 +; 0 <<)
% Number of symbols : 7 ( 4 usr; 3 con; 0-3 aty)
% Number of variables : 108 ( 0 ^; 99 !; 9 ?; 108 :)
% Comments :
%------------------------------------------------------------------------------
thf(zero_decl,type,
zero: $i ).
thf(s_decl,type,
s: $i > $i ).
thf(ite_decl,type,
ite: $o > $i > $i > $i ).
thf(h_decl,type,
h: $i > $i ).
thf(1,conjecture,
( ( ! [A: $o,B: $i,C: $i] :
( A
=> ( ( ite @ A @ B @ C )
= B ) )
& ! [A: $o,B: $i,C: $i] :
( ~ A
=> ( ( ite @ A @ B @ C )
= C ) )
& ! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',n10) ).
thf(2,negated_conjecture,
~ ( ( ! [A: $o,B: $i,C: $i] :
( A
=> ( ( ite @ A @ B @ C )
= B ) )
& ! [A: $o,B: $i,C: $i] :
( ~ A
=> ( ( ite @ A @ B @ C )
= C ) )
& ! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(3,plain,
~ ( ( ! [A: $o,B: $i,C: $i] :
( A
=> ( ( ite @ A @ B @ C )
= B ) )
& ! [A: $o,B: $i,C: $i] :
( ~ A
=> ( ( ite @ A @ B @ C )
= C ) )
& ! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(10,plain,
~ ( ( ! [A: $i,B: $i] :
( $false
=> ( ( ite @ $false @ A @ B )
= A ) )
& ! [A: $i,B: $i] :
( ~ $false
=> ( ( ite @ $false @ A @ B )
= B ) )
& ! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero ) ) ),
inference(instance,[status(thm)],[3]) ).
thf(11,plain,
~ ( ( ! [A: $i,B: $i] :
( ( ite @ $false @ A @ B )
= B )
& ! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero ) ) ),
inference(simp,[status(thm)],[10]) ).
thf(24,plain,
! [D: $i > $i,C: $i,B: $i,A: $i] :
( ( ( ite @ $false @ A @ B )
= B )
& ( ( h @ C )
= ( ite @ ( C = zero ) @ ( s @ zero ) @ zero ) )
& ( ( ( D @ zero )
!= ( s @ zero ) )
| ( ( D @ ( s @ zero ) )
!= zero ) ) ),
inference(cnf,[status(esa)],[11]) ).
thf(25,plain,
! [B: $i,A: $i] :
( ( ite @ $false @ A @ B )
= B ),
inference(cnfConj,[status(thm)],[24]) ).
thf(28,plain,
! [B: $i,A: $i] :
( ( ite @ $false @ A @ B )
= B ),
inference(lifteq,[status(thm)],[25]) ).
thf(4,plain,
~ ( ( ! [A: $i,B: $i] :
( $true
=> ( ( ite @ $true @ A @ B )
= A ) )
& ! [A: $i,B: $i] :
( ~ $true
=> ( ( ite @ $true @ A @ B )
= B ) )
& ! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero ) ) ),
inference(instance,[status(thm)],[3]) ).
thf(5,plain,
~ ( ( ! [A: $i,B: $i] :
( ( ite @ $true @ A @ B )
= A )
& ! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero ) ) ),
inference(simp,[status(thm)],[4]) ).
thf(12,plain,
! [D: $i > $i,C: $i,B: $i,A: $i] :
( ( ( ite @ $true @ A @ B )
= A )
& ( ( h @ C )
= ( ite @ ( C = zero ) @ ( s @ zero ) @ zero ) )
& ( ( ( D @ zero )
!= ( s @ zero ) )
| ( ( D @ ( s @ zero ) )
!= zero ) ) ),
inference(cnf,[status(esa)],[5]) ).
thf(14,plain,
! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ),
inference(cnfConj,[status(thm)],[12]) ).
thf(29,plain,
! [A: $i] :
( ( ite @ ( A = zero ) @ ( s @ zero ) @ zero )
= ( h @ A ) ),
inference(lifteq,[status(thm)],[14]) ).
thf(30,plain,
! [A: $i] :
( ( ite @ ( A = zero ) @ ( s @ zero ) @ zero )
= ( h @ A ) ),
inference(simp,[status(thm)],[29]) ).
thf(75,plain,
! [C: $i,B: $i,A: $i] :
( ( B
= ( h @ C ) )
| ( ( ite @ $false @ A @ B )
!= ( ite @ ( C = zero ) @ ( s @ zero ) @ zero ) ) ),
inference(paramod_ordered,[status(thm)],[28,30]) ).
thf(87,plain,
! [C: $i,B: $i,A: $i] :
( ( B
= ( h @ C ) )
| ( C = zero )
| ( A
!= ( s @ zero ) )
| ( B != zero ) ),
inference(simp,[status(thm)],[75]) ).
thf(96,plain,
! [C: $i,B: $i,A: $i] :
( ( C = zero )
| ( B
= ( h @ C ) )
| ( A
!= ( s @ zero ) )
| ( B != zero ) ),
inference(lifteq,[status(thm)],[87]) ).
thf(97,plain,
! [A: $i] :
( ( A = zero )
| ( ( h @ A )
= zero ) ),
inference(simp,[status(thm)],[96]) ).
thf(13,plain,
! [B: $i,A: $i] :
( ( ite @ $true @ A @ B )
= A ),
inference(cnfConj,[status(thm)],[12]) ).
thf(33,plain,
! [B: $i,A: $i] :
( ( ite @ $true @ A @ B )
= A ),
inference(lifteq,[status(thm)],[13]) ).
thf(76,plain,
! [C: $i,B: $i,A: $i] :
( ( ( h @ A )
= B )
| ( ( ite @ ( A = zero ) @ ( s @ zero ) @ zero )
!= ( ite @ $true @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[30,33]) ).
thf(82,plain,
! [C: $i,B: $i,A: $i] :
( ( ( h @ A )
= B )
| ( A != zero )
| ( ( s @ zero )
!= B )
| ( zero != C ) ),
inference(simp,[status(thm)],[76]) ).
thf(90,plain,
! [C: $i,B: $i,A: $i] :
( ( A != zero )
| ( ( h @ A )
= B )
| ( ( s @ zero )
!= B )
| ( zero != C ) ),
inference(lifteq,[status(thm)],[82]) ).
thf(91,plain,
( ( h @ zero )
= ( s @ zero ) ),
inference(simp,[status(thm)],[90]) ).
thf(6,plain,
~ ( ( ! [A: $i,B: $i] :
( $true
=> ( ( ite @ $true @ A @ B )
= A ) )
& ! [A: $i,B: $i] :
( ~ $false
=> ( ( ite @ $false @ A @ B )
= B ) )
& ! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero ) ) ),
inference(instance,[status(thm)],[3]) ).
thf(7,plain,
~ ( ( ! [A: $i,B: $i] :
( ( ite @ $true @ A @ B )
= A )
& ! [A: $i,B: $i] :
( ( ite @ $false @ A @ B )
= B )
& ! [A: $i] :
( ( h @ A )
= ( ite @ ( A = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero ) ) ),
inference(simp,[status(thm)],[6]) ).
thf(16,plain,
! [F: $i > $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( ite @ $true @ A @ B )
= A )
& ( ( ite @ $false @ C @ D )
= D )
& ( ( h @ E )
= ( ite @ ( E = zero ) @ ( s @ zero ) @ zero ) )
& ( ( ( F @ zero )
!= ( s @ zero ) )
| ( ( F @ ( s @ zero ) )
!= zero ) ) ),
inference(cnf,[status(esa)],[7]) ).
thf(20,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ zero ) )
!= zero ) ),
inference(cnfConj,[status(thm)],[16]) ).
thf(31,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ zero ) )
!= zero ) ),
inference(lifteq,[status(thm)],[20]) ).
thf(32,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ zero ) )
!= zero ) ),
inference(simp,[status(thm)],[31]) ).
thf(102,plain,
! [A: $i > $i] :
( ( ( s @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ zero ) )
!= zero )
| ( ( h @ zero )
!= ( A @ zero ) ) ),
inference(paramod_ordered,[status(thm)],[91,32]) ).
thf(103,plain,
! [A: $i > $i] :
( ( ( A @ ( s @ zero ) )
!= zero )
| ( ( h @ zero )
!= ( A @ zero ) ) ),
inference(simp,[status(thm)],[102]) ).
thf(106,plain,
( ( h @ ( s @ zero ) )
!= zero ),
inference(pre_uni,[status(thm)],[103:[bind(A,$thf( h ))]]) ).
thf(122,plain,
! [A: $i] :
( ( A = zero )
| ( zero != zero )
| ( ( h @ A )
!= ( h @ ( s @ zero ) ) ) ),
inference(paramod_ordered,[status(thm)],[97,106]) ).
thf(123,plain,
! [A: $i] :
( ( A = zero )
| ( ( h @ A )
!= ( h @ ( s @ zero ) ) ) ),
inference(simp,[status(thm)],[122]) ).
thf(124,plain,
( ( s @ zero )
= zero ),
inference(pattern_uni,[status(thm)],[123:[bind(A,$thf( s @ zero ))]]) ).
thf(62,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ zero ) )
!= ( s @ zero ) )
| ( ( A @ zero )
!= zero ) ),
inference(eqfactor_ordered,[status(thm)],[32]) ).
thf(67,plain,
( ( s @ zero )
!= zero ),
inference(pre_uni,[status(thm)],[62:[bind(A,$thf( ^ [B: $i] : B ))]]) ).
thf(138,plain,
$false,
inference(simplifyReflect,[status(thm)],[124,67]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : NUN023^2 : TPTP v9.3.1. Released v6.4.0.
% 0.00/0.07 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.09/0.20 % Computer : n005.cluster.edu
% 0.09/0.20 % Model : x86_64 x86_64
% 0.09/0.20 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.20 % Memory : 8046.5625MB
% 0.09/0.20 % OS : Linux 6.8.0-71-generic
% 0.09/0.20 % CPULimit : 300
% 0.09/0.20 % WCLimit : 300
% 0.09/0.20 % DateTime : Tue Sep 29 13:06:01 UTC 2026
% 0.09/0.20 % CPUTime :
% 0.09/0.20 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.02/0.80 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.22/0.99 % [INFO] Parsing done (184ms).
% 1.22/1.00 % [INFO] Running in sequential loop mode.
% 2.02/1.40 % [INFO] eprover registered as external prover.
% 2.02/1.41 % [INFO] Scanning for conjecture ...
% 2.21/1.51 % [INFO] Found a conjecture (or negated_conjecture) and 0 axioms. Running axiom selection ...
% 2.21/1.55 % [INFO] Axiom selection finished. Selected 0 axioms (removed 0 axioms).
% 2.21/1.55 % [INFO] Problem is higher-order (TPTP THF).
% 2.21/1.56 % [INFO] Type checking passed.
% 2.21/1.56 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 5.20/2.70 % [INFO] Killing All external provers ...
% 5.20/2.70 % Time passed: 2363ms (effective reasoning time: 1695ms)
% 5.20/2.70 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 5.20/2.71 % Axioms used in derivation (0):
% 5.20/2.71 % No. of inferences in proof: 39
% 5.20/2.71 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 2363 ms resp. 1695 ms w/o parsing
% 5.20/2.77 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 5.20/2.77 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------