%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : NUN025^2 : TPTP v9.3.1. Released v6.4.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n008.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 08:19:49 AM UTC 2026
% Result : Theorem 5.52s 2.90s
% Output : Refutation 5.52s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 1
% Syntax : Number of formulae : 42 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 201 ( 170 equ; 0 cnn)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 629 ( 81 ~; 32 |; 66 &; 426 @)
% ( 0 <=>; 24 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 21 ( 21 >; 0 *; 0 +; 0 <<)
% Number of symbols : 7 ( 4 usr; 3 con; 0-3 aty)
% Number of variables : 130 ( 0 ^; 115 !; 15 ?; 130 :)
% 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] :
( ( s @ A )
!= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ zero ) ) ) ),
file('/export/starexec/sandbox/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] :
( ( s @ A )
!= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ 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] :
( ( s @ A )
!= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ 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] :
( ( s @ A )
!= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ zero ) ) ) ),
inference(instance,[status(thm)],[3]) ).
thf(11,plain,
~ ( ( ! [A: $i,B: $i] :
( ( ite @ $false @ A @ B )
= B )
& ! [A: $i] :
( ( s @ A )
!= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ zero ) ) ) ),
inference(simp,[status(thm)],[10]) ).
thf(30,plain,
~ ( ( ! [A: $i,B: $i] :
( ( ite @ $false @ A @ B )
= B )
& ~ ? [A: $i] :
( ( s @ A )
= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ zero ) ) ) ),
inference(miniscope,[status(thm)],[11]) ).
thf(31,plain,
! [E: $i > $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( ite @ $false @ A @ B )
= B )
& ( ( s @ C )
!= C )
& ( ( h @ D )
= ( ite
@ ( D
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) )
& ( ( ( E @ zero )
!= ( s @ zero ) )
| ( ( E @ ( s @ zero ) )
!= zero )
| ( ( E @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) ) ) ),
inference(cnf,[status(esa)],[30]) ).
thf(35,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ zero ) )
!= zero )
| ( ( A @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) ) ),
inference(cnfConj,[status(thm)],[31]) ).
thf(39,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ zero ) )
!= zero )
| ( ( A @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) ) ),
inference(lifteq,[status(thm)],[35]) ).
thf(40,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ zero ) )
!= zero )
| ( ( A @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) ) ),
inference(simp,[status(thm)],[39]) ).
thf(65,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ zero ) )
!= zero )
| ( ( A @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) )
| ( ( s @ zero )
!= ( A @ zero ) ) ),
inference(eqfactor_ordered,[status(thm)],[40]) ).
thf(68,plain,
( ( ( s @ zero )
!= ( s @ zero ) )
| ( ( s @ zero )
!= zero ) ),
inference(pre_uni,[status(thm)],[65:[bind(A,$thf( ^ [B: $i] : ( s @ zero ) ))]]) ).
thf(69,plain,
( ( s @ zero )
!= zero ),
inference(simp,[status(thm)],[68]) ).
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] :
( ( s @ A )
!= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ 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] :
( ( s @ A )
!= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ zero ) ) ) ),
inference(simp,[status(thm)],[6]) ).
thf(18,plain,
~ ( ( ! [A: $i,B: $i] :
( ( ite @ $true @ A @ B )
= A )
& ! [A: $i,B: $i] :
( ( ite @ $false @ A @ B )
= B )
& ~ ? [A: $i] :
( ( s @ A )
= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ zero ) ) ) ),
inference(miniscope,[status(thm)],[7]) ).
thf(19,plain,
! [G: $i > $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( ite @ $true @ A @ B )
= A )
& ( ( ite @ $false @ C @ D )
= D )
& ( ( s @ E )
!= E )
& ( ( h @ F )
= ( ite
@ ( F
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) )
& ( ( ( G @ zero )
!= ( s @ zero ) )
| ( ( G @ ( s @ zero ) )
!= zero )
| ( ( G @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) ) ) ),
inference(cnf,[status(esa)],[18]) ).
thf(24,plain,
! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ),
inference(cnfConj,[status(thm)],[19]) ).
thf(36,plain,
! [A: $i] :
( ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) )
= ( h @ A ) ),
inference(lifteq,[status(thm)],[24]) ).
thf(37,plain,
! [A: $i] :
( ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) )
= ( h @ A ) ),
inference(simp,[status(thm)],[36]) ).
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] :
( ( s @ A )
!= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ zero ) ) ) ),
inference(instance,[status(thm)],[3]) ).
thf(5,plain,
~ ( ( ! [A: $i,B: $i] :
( ( ite @ $true @ A @ B )
= A )
& ! [A: $i] :
( ( s @ A )
!= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ zero ) ) ) ),
inference(simp,[status(thm)],[4]) ).
thf(12,plain,
~ ( ( ! [A: $i,B: $i] :
( ( ite @ $true @ A @ B )
= A )
& ~ ? [A: $i] :
( ( s @ A )
= A )
& ! [A: $i] :
( ( h @ A )
= ( ite
@ ( A
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) )
=> ? [A: $i > $i] :
( ( ( A @ zero )
= ( s @ zero ) )
& ( ( A @ ( s @ zero ) )
= zero )
& ( ( A @ ( s @ ( s @ zero ) ) )
= ( s @ zero ) ) ) ),
inference(miniscope,[status(thm)],[5]) ).
thf(13,plain,
! [E: $i > $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( ite @ $true @ A @ B )
= A )
& ( ( s @ C )
!= C )
& ( ( h @ D )
= ( ite
@ ( D
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) )
& ( ( ( E @ zero )
!= ( s @ zero ) )
| ( ( E @ ( s @ zero ) )
!= zero )
| ( ( E @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) ) ) ),
inference(cnf,[status(esa)],[12]) ).
thf(14,plain,
! [B: $i,A: $i] :
( ( ite @ $true @ A @ B )
= A ),
inference(cnfConj,[status(thm)],[13]) ).
thf(38,plain,
! [B: $i,A: $i] :
( ( ite @ $true @ A @ B )
= A ),
inference(lifteq,[status(thm)],[14]) ).
thf(23,plain,
! [A: $i] :
( ( s @ A )
!= A ),
inference(cnfConj,[status(thm)],[19]) ).
thf(43,plain,
! [A: $i] :
( ( s @ A )
!= A ),
inference(lifteq,[status(thm)],[23]) ).
thf(44,plain,
! [A: $i] :
( ( s @ A )
!= A ),
inference(simp,[status(thm)],[43]) ).
thf(20,plain,
! [B: $i,A: $i] :
( ( ite @ $false @ A @ B )
= B ),
inference(cnfConj,[status(thm)],[19]) ).
thf(47,plain,
! [B: $i,A: $i] :
( ( ite @ $false @ A @ B )
= B ),
inference(lifteq,[status(thm)],[20]) ).
thf(48,plain,
! [B: $i,A: $i] :
( ( ite @ $false @ A @ B )
= B ),
inference(simp,[status(thm)],[47]) ).
thf(97,plain,
! [C: $i,B: $i,A: $i] :
( ( A
= ( h @ C ) )
| ( ( ite @ $true @ A @ B )
!= ( ite
@ ( C
= ( s @ zero ) )
@ zero
@ ( s @ zero ) ) ) ),
inference(paramod_ordered,[status(thm)],[38,37]) ).
thf(108,plain,
! [C: $i,B: $i,A: $i] :
( ( A
= ( h @ C ) )
| ( C
!= ( s @ zero ) )
| ( A != zero )
| ( B
!= ( s @ zero ) ) ),
inference(simp,[status(thm)],[97]) ).
thf(121,plain,
! [C: $i,B: $i,A: $i] :
( ( C
!= ( s @ zero ) )
| ( A
= ( h @ C ) )
| ( A != zero )
| ( B
!= ( s @ zero ) ) ),
inference(lifteq,[status(thm)],[108]) ).
thf(122,plain,
( ( h @ ( s @ zero ) )
= zero ),
inference(simp,[status(thm)],[121]) ).
thf(134,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( zero != zero )
| ( ( A @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) )
| ( ( h @ ( s @ zero ) )
!= ( A @ ( s @ zero ) ) ) ),
inference(paramod_ordered,[status(thm)],[122,40]) ).
thf(135,plain,
! [A: $i > $i] :
( ( ( A @ zero )
!= ( s @ zero ) )
| ( ( A @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) )
| ( ( h @ ( s @ zero ) )
!= ( A @ ( s @ zero ) ) ) ),
inference(simp,[status(thm)],[134]) ).
thf(139,plain,
( ( ( h @ zero )
!= ( s @ zero ) )
| ( ( h @ ( s @ ( s @ zero ) ) )
!= ( s @ zero ) ) ),
inference(pre_uni,[status(thm)],[135:[bind(A,$thf( h ))]]) ).
thf(150,plain,
( ( ( h @ zero )
!= ( s @ zero ) )
| ( ( h @ ( s @ ( s @ zero ) ) )
!= ( h @ zero ) )
| ( ( s @ zero )
!= ( s @ zero ) ) ),
inference(eqfactor_ordered,[status(thm)],[139]) ).
thf(154,plain,
( ( ( h @ zero )
!= ( s @ zero ) )
| ( ( s @ ( s @ zero ) )
!= zero ) ),
inference(simp,[status(thm)],[150]) ).
thf(198,plain,
$false,
inference(e,[status(thm)],[69,37,38,44,3,48,154,122,40,139]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : NUN025^2 : TPTP v9.3.1. Released v6.4.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.10/0.21 % Computer : n008.cluster.edu
% 0.10/0.21 % Model : x86_64 x86_64
% 0.10/0.21 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.21 % Memory : 8046.5625MB
% 0.10/0.21 % OS : Linux 6.8.0-71-generic
% 0.10/0.21 % CPULimit : 300
% 0.10/0.21 % WCLimit : 300
% 0.10/0.21 % DateTime : Tue Sep 29 13:06:54 UTC 2026
% 0.10/0.21 % CPUTime :
% 0.10/0.21 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 1.01/0.80 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.18/1.00 % [INFO] Parsing done (192ms).
% 1.18/1.01 % [INFO] Running in sequential loop mode.
% 1.92/1.41 % [INFO] eprover registered as external prover.
% 1.92/1.42 % [INFO] Scanning for conjecture ...
% 2.10/1.52 % [INFO] Found a conjecture (or negated_conjecture) and 0 axioms. Running axiom selection ...
% 2.10/1.56 % [INFO] Axiom selection finished. Selected 0 axioms (removed 0 axioms).
% 2.10/1.56 % [INFO] Problem is higher-order (TPTP THF).
% 2.10/1.56 % [INFO] Type checking passed.
% 2.10/1.57 % [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 ...
% 5.52/2.90 % External prover 'e' found a proof!
% 5.52/2.90 % [INFO] Killing All external provers ...
% 5.52/2.90 % Time passed: 2543ms (effective reasoning time: 1882ms)
% 5.52/2.90 % 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)>
% 5.52/2.90 % Axioms used in derivation (0):
% 5.52/2.90 % No. of inferences in proof: 42
% 5.52/2.90 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 2543 ms resp. 1882 ms w/o parsing
% 5.52/2.97 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 5.52/2.97 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------