%------------------------------------------------------------------------------
% File : LEO-II---2.3.1
% Problem : NUN024^2 : TPTP v9.3.1. Released v6.4.0.
% Transfm : none
% Format : tptp:raw
% Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox/solver/bin/eprover /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n013.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 Oct 7 10:35:55 AM UTC 2026
% Result : Theorem 0.09s 0.19s
% Output : CNFRefutation 0.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 1
% Syntax : Number of formulae : 61 ( 34 unt; 0 typ; 0 def)
% Number of atoms : 322 ( 195 equ; 0 cnn)
% Maximal formula atoms : 8 ( 5 avg)
% Number of connectives : 379 ( 45 ~; 47 |; 22 &; 254 @)
% ( 0 <=>; 11 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 2 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 13 ( 13 >; 0 *; 0 +; 0 <<)
% Number of symbols : 7 ( 4 usr; 3 con; 0-3 aty)
% Number of variables : 134 ( 0 ^; 129 !; 5 ?; 134 :)
% Comments :
%------------------------------------------------------------------------------
thf(tp_h,type,
h: $i > $i ).
thf(tp_ite,type,
ite: $o > $i > $i > $i ).
thf(tp_s,type,
s: $i > $i ).
thf(tp_zero,type,
zero: $i ).
thf(1,conjecture,
( ( ! [X100: $o,U: $i,V: $i] :
( X100
=> ( ( ite @ X100 @ U @ V )
= U ) )
& ! [X100: $o,U: $i,V: $i] :
( ~ X100
=> ( ( ite @ X100 @ U @ V )
= V ) )
& ! [X: $i] :
( ( s @ X )
!= zero )
& ! [X: $i] :
( ( s @ X )
!= X )
& ! [X: $i] :
( ( h @ X )
= ( ite @ ( X = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [H: $i > $i] :
( ( ( H @ zero )
= ( s @ zero ) )
& ( ( H @ ( s @ zero ) )
= zero )
& ( ( H @ ( s @ ( s @ zero ) ) )
= zero ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',n10) ).
thf(2,negated_conjecture,
( ( ( ! [X100: $o,U: $i,V: $i] :
( X100
=> ( ( ite @ X100 @ U @ V )
= U ) )
& ! [X100: $o,U: $i,V: $i] :
( ~ X100
=> ( ( ite @ X100 @ U @ V )
= V ) )
& ! [X: $i] :
( ( s @ X )
!= zero )
& ! [X: $i] :
( ( s @ X )
!= X )
& ! [X: $i] :
( ( h @ X )
= ( ite @ ( X = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [H: $i > $i] :
( ( ( H @ zero )
= ( s @ zero ) )
& ( ( H @ ( s @ zero ) )
= zero )
& ( ( H @ ( s @ ( s @ zero ) ) )
= zero ) ) )
= $false ),
inference(negate_conjecture,[status(cth)],[1]) ).
thf(3,plain,
( ( ( ! [X100: $o,U: $i,V: $i] :
( X100
=> ( ( ite @ X100 @ U @ V )
= U ) )
& ! [X100: $o,U: $i,V: $i] :
( ~ X100
=> ( ( ite @ X100 @ U @ V )
= V ) )
& ! [X: $i] :
( ( s @ X )
!= zero )
& ! [X: $i] :
( ( s @ X )
!= X )
& ! [X: $i] :
( ( h @ X )
= ( ite @ ( X = zero ) @ ( s @ zero ) @ zero ) ) )
=> ? [H: $i > $i] :
( ( ( H @ zero )
= ( s @ zero ) )
& ( ( H @ ( s @ zero ) )
= zero )
& ( ( H @ ( s @ ( s @ zero ) ) )
= zero ) ) )
= $false ),
inference(unfold_def,[status(thm)],[2]) ).
thf(4,plain,
( ( ! [X100: $o,U: $i,V: $i] :
( X100
=> ( ( ite @ X100 @ U @ V )
= U ) ) )
= $true ),
inference(standard_cnf,[status(thm)],[3]) ).
thf(5,plain,
( ( ! [X100: $o,U: $i,V: $i] :
( ~ X100
=> ( ( ite @ X100 @ U @ V )
= V ) ) )
= $true ),
inference(standard_cnf,[status(thm)],[3]) ).
thf(6,plain,
( ( ! [X: $i] :
( ( s @ X )
!= zero ) )
= $true ),
inference(standard_cnf,[status(thm)],[3]) ).
thf(7,plain,
( ( ! [X: $i] :
( ( s @ X )
!= X ) )
= $true ),
inference(standard_cnf,[status(thm)],[3]) ).
thf(8,plain,
( ( ! [X: $i] :
( ( h @ X )
= ( ite @ ( X = zero ) @ ( s @ zero ) @ zero ) ) )
= $true ),
inference(standard_cnf,[status(thm)],[3]) ).
thf(9,plain,
( ( ? [H: $i > $i] :
( ( ( H @ zero )
= ( s @ zero ) )
& ( ( H @ ( s @ zero ) )
= zero )
& ( ( H @ ( s @ ( s @ zero ) ) )
= zero ) ) )
= $false ),
inference(standard_cnf,[status(thm)],[3]) ).
thf(10,plain,
( ( ~ ? [H: $i > $i] :
( ( ( H @ zero )
= ( s @ zero ) )
& ( ( H @ ( s @ zero ) )
= zero )
& ( ( H @ ( s @ ( s @ zero ) ) )
= zero ) ) )
= $true ),
inference(polarity_switch,[status(thm)],[9]) ).
thf(11,plain,
( ( ! [H: $i > $i] :
( ( ( H @ zero )
!= ( s @ zero ) )
| ( ( H @ ( s @ zero ) )
!= zero )
| ( ( H @ ( s @ ( s @ zero ) ) )
!= zero ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[10]) ).
thf(12,plain,
( ( ! [X100: $o] :
( ~ X100
| ! [U: $i,V: $i] :
( ( ite @ X100 @ U @ V )
= U ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[4]) ).
thf(13,plain,
( ( ! [X100: $o] :
( X100
| ! [U: $i,V: $i] :
( ( ite @ X100 @ U @ V )
= V ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[5]) ).
thf(14,plain,
( ( ! [X: $i] :
( ( s @ X )
!= zero ) )
= $true ),
inference(extcnf_combined,[status(esa)],[6]) ).
thf(15,plain,
( ( ! [X: $i] :
( ( s @ X )
!= X ) )
= $true ),
inference(extcnf_combined,[status(esa)],[7]) ).
thf(16,plain,
( ( ! [X: $i] :
( ( h @ X )
= ( ite @ ( X = zero ) @ ( s @ zero ) @ zero ) ) )
= $true ),
inference(copy,[status(thm)],[8]) ).
thf(17,plain,
( ( ! [X: $i] :
( ( s @ X )
!= X ) )
= $true ),
inference(copy,[status(thm)],[15]) ).
thf(18,plain,
( ( ! [X: $i] :
( ( s @ X )
!= zero ) )
= $true ),
inference(copy,[status(thm)],[14]) ).
thf(19,plain,
( ( ! [X100: $o] :
( X100
| ! [U: $i,V: $i] :
( ( ite @ X100 @ U @ V )
= V ) ) )
= $true ),
inference(copy,[status(thm)],[13]) ).
thf(20,plain,
( ( ! [X100: $o] :
( ~ X100
| ! [U: $i,V: $i] :
( ( ite @ X100 @ U @ V )
= U ) ) )
= $true ),
inference(copy,[status(thm)],[12]) ).
thf(21,plain,
( ( ! [H: $i > $i] :
( ( ( H @ zero )
!= ( s @ zero ) )
| ( ( H @ ( s @ zero ) )
!= zero )
| ( ( H @ ( s @ ( s @ zero ) ) )
!= zero ) ) )
= $true ),
inference(copy,[status(thm)],[11]) ).
thf(22,plain,
! [SV1: $i] :
( ( ( h @ SV1 )
= ( ite @ ( SV1 = zero ) @ ( s @ zero ) @ zero ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[16]) ).
thf(23,plain,
! [SV2: $i] :
( ( ( s @ SV2 )
!= SV2 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[17]) ).
thf(24,plain,
! [SV3: $i] :
( ( ( s @ SV3 )
!= zero )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[18]) ).
thf(25,plain,
! [SV4: $o] :
( ( SV4
| ! [SY10: $i,SY11: $i] :
( ( ite @ SV4 @ SY10 @ SY11 )
= SY11 ) )
= $true ),
inference(extcnf_forall_special_pos,[status(thm)],[19]) ).
thf(26,plain,
( ( $true
| ! [SX0: $i,SX1: $i] :
( ( ite @ $true @ SX0 @ SX1 )
= SX1 ) )
= $true ),
inference(extcnf_forall_special_pos,[status(thm)],[19]) ).
thf(27,plain,
( ( $false
| ! [SX0: $i,SX1: $i] :
( ( ite @ $false @ SX0 @ SX1 )
= SX1 ) )
= $true ),
inference(extcnf_forall_special_pos,[status(thm)],[19]) ).
thf(28,plain,
! [SV5: $o] :
( ( ~ SV5
| ! [SY12: $i,SY13: $i] :
( ( ite @ SV5 @ SY12 @ SY13 )
= SY12 ) )
= $true ),
inference(extcnf_forall_special_pos,[status(thm)],[20]) ).
thf(29,plain,
( ( ~ $true
| ! [SX0: $i,SX1: $i] :
( ( ite @ $true @ SX0 @ SX1 )
= SX0 ) )
= $true ),
inference(extcnf_forall_special_pos,[status(thm)],[20]) ).
thf(30,plain,
( ( ~ $false
| ! [SX0: $i,SX1: $i] :
( ( ite @ $false @ SX0 @ SX1 )
= SX0 ) )
= $true ),
inference(extcnf_forall_special_pos,[status(thm)],[20]) ).
thf(31,plain,
! [SV6: $i > $i] :
( ( ( ( SV6 @ zero )
!= ( s @ zero ) )
| ( ( SV6 @ ( s @ zero ) )
!= zero )
| ( ( SV6 @ ( s @ ( s @ zero ) ) )
!= zero ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[21]) ).
thf(32,plain,
! [SV2: $i] :
( ( ( s @ SV2 )
= SV2 )
= $false ),
inference(extcnf_not_pos,[status(thm)],[23]) ).
thf(33,plain,
! [SV3: $i] :
( ( ( s @ SV3 )
= zero )
= $false ),
inference(extcnf_not_pos,[status(thm)],[24]) ).
thf(34,plain,
! [SV4: $o] :
( ( SV4 = $true )
| ( ( ! [SY10: $i,SY11: $i] :
( ( ite @ SV4 @ SY10 @ SY11 )
= SY11 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[25]) ).
thf(35,plain,
( ( $true = $true )
| ( ( ! [SX0: $i,SX1: $i] :
( ( ite @ $true @ SX0 @ SX1 )
= SX1 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[26]) ).
thf(36,plain,
( ( $false = $true )
| ( ( ! [SX0: $i,SX1: $i] :
( ( ite @ $false @ SX0 @ SX1 )
= SX1 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[27]) ).
thf(37,plain,
! [SV5: $o] :
( ( ( ~ SV5 ) = $true )
| ( ( ! [SY12: $i,SY13: $i] :
( ( ite @ SV5 @ SY12 @ SY13 )
= SY12 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[28]) ).
thf(38,plain,
( ( ( ~ $true ) = $true )
| ( ( ! [SX0: $i,SX1: $i] :
( ( ite @ $true @ SX0 @ SX1 )
= SX0 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[29]) ).
thf(39,plain,
( ( ( ~ $false ) = $true )
| ( ( ! [SX0: $i,SX1: $i] :
( ( ite @ $false @ SX0 @ SX1 )
= SX0 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[30]) ).
thf(40,plain,
! [SV6: $i > $i] :
( ( ( ( ( SV6 @ zero )
!= ( s @ zero ) )
| ( ( SV6 @ ( s @ zero ) )
!= zero ) )
= $true )
| ( ( ( SV6 @ ( s @ ( s @ zero ) ) )
!= zero )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[31]) ).
thf(41,plain,
! [SV7: $i,SV4: $o] :
( ( ( ! [SY14: $i] :
( ( ite @ SV4 @ SV7 @ SY14 )
= SY14 ) )
= $true )
| ( SV4 = $true ) ),
inference(extcnf_forall_pos,[status(thm)],[34]) ).
thf(42,plain,
! [SV8: $i] :
( ( ( ! [SY15: $i] :
( ( ite @ $true @ SV8 @ SY15 )
= SY15 ) )
= $true )
| ( $true = $true ) ),
inference(extcnf_forall_pos,[status(thm)],[35]) ).
thf(43,plain,
! [SV9: $i] :
( ( ( ! [SY16: $i] :
( ( ite @ $false @ SV9 @ SY16 )
= SY16 ) )
= $true )
| ( $false = $true ) ),
inference(extcnf_forall_pos,[status(thm)],[36]) ).
thf(44,plain,
! [SV5: $o] :
( ( SV5 = $false )
| ( ( ! [SY12: $i,SY13: $i] :
( ( ite @ SV5 @ SY12 @ SY13 )
= SY12 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[37]) ).
thf(45,plain,
( ( $true = $false )
| ( ( ! [SX0: $i,SX1: $i] :
( ( ite @ $true @ SX0 @ SX1 )
= SX0 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[38]) ).
thf(46,plain,
( ( $false = $false )
| ( ( ! [SX0: $i,SX1: $i] :
( ( ite @ $false @ SX0 @ SX1 )
= SX0 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[39]) ).
thf(47,plain,
! [SV6: $i > $i] :
( ( ( ( SV6 @ zero )
!= ( s @ zero ) )
= $true )
| ( ( ( SV6 @ ( s @ zero ) )
!= zero )
= $true )
| ( ( ( SV6 @ ( s @ ( s @ zero ) ) )
!= zero )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[40]) ).
thf(48,plain,
! [SV10: $i,SV7: $i,SV4: $o] :
( ( ( ( ite @ SV4 @ SV7 @ SV10 )
= SV10 )
= $true )
| ( SV4 = $true ) ),
inference(extcnf_forall_pos,[status(thm)],[41]) ).
thf(49,plain,
! [SV11: $i,SV8: $i] :
( ( ( ( ite @ $true @ SV8 @ SV11 )
= SV11 )
= $true )
| ( $true = $true ) ),
inference(extcnf_forall_pos,[status(thm)],[42]) ).
thf(50,plain,
! [SV12: $i,SV9: $i] :
( ( ( ( ite @ $false @ SV9 @ SV12 )
= SV12 )
= $true )
| ( $false = $true ) ),
inference(extcnf_forall_pos,[status(thm)],[43]) ).
thf(51,plain,
! [SV13: $i,SV5: $o] :
( ( ( ! [SY17: $i] :
( ( ite @ SV5 @ SV13 @ SY17 )
= SV13 ) )
= $true )
| ( SV5 = $false ) ),
inference(extcnf_forall_pos,[status(thm)],[44]) ).
thf(52,plain,
! [SV14: $i] :
( ( ( ! [SY18: $i] :
( ( ite @ $true @ SV14 @ SY18 )
= SV14 ) )
= $true )
| ( $true = $false ) ),
inference(extcnf_forall_pos,[status(thm)],[45]) ).
thf(53,plain,
! [SV15: $i] :
( ( ( ! [SY19: $i] :
( ( ite @ $false @ SV15 @ SY19 )
= SV15 ) )
= $true )
| ( $false = $false ) ),
inference(extcnf_forall_pos,[status(thm)],[46]) ).
thf(54,plain,
! [SV6: $i > $i] :
( ( ( ( SV6 @ zero )
= ( s @ zero ) )
= $false )
| ( ( ( SV6 @ ( s @ zero ) )
!= zero )
= $true )
| ( ( ( SV6 @ ( s @ ( s @ zero ) ) )
!= zero )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[47]) ).
thf(55,plain,
! [SV16: $i,SV13: $i,SV5: $o] :
( ( ( ( ite @ SV5 @ SV13 @ SV16 )
= SV13 )
= $true )
| ( SV5 = $false ) ),
inference(extcnf_forall_pos,[status(thm)],[51]) ).
thf(56,plain,
! [SV17: $i,SV14: $i] :
( ( ( ( ite @ $true @ SV14 @ SV17 )
= SV14 )
= $true )
| ( $true = $false ) ),
inference(extcnf_forall_pos,[status(thm)],[52]) ).
thf(57,plain,
! [SV18: $i,SV15: $i] :
( ( ( ( ite @ $false @ SV15 @ SV18 )
= SV15 )
= $true )
| ( $false = $false ) ),
inference(extcnf_forall_pos,[status(thm)],[53]) ).
thf(58,plain,
! [SV6: $i > $i] :
( ( ( ( SV6 @ ( s @ zero ) )
= zero )
= $false )
| ( ( ( SV6 @ zero )
= ( s @ zero ) )
= $false )
| ( ( ( SV6 @ ( s @ ( s @ zero ) ) )
!= zero )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[54]) ).
thf(59,plain,
! [SV6: $i > $i] :
( ( ( ( SV6 @ ( s @ ( s @ zero ) ) )
= zero )
= $false )
| ( ( ( SV6 @ zero )
= ( s @ zero ) )
= $false )
| ( ( ( SV6 @ ( s @ zero ) )
= zero )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[58]) ).
thf(60,plain,
$false = $true,
inference(fo_atp_e,[status(thm)],[22,59,57,56,55,50,49,48,33,32]) ).
thf(61,plain,
$false,
inference(solved_all_splits,[solved_all_splits(join,[])],[60]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : NUN024^2 : TPTP v9.3.1. Released v6.4.0.
% 0.00/0.03 % Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox/solver/bin/eprover /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.15 % Computer : n013.cluster.edu
% 0.09/0.15 % Model : x86_64 x86_64
% 0.09/0.15 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.15 % Memory : 8046.5625MB
% 0.09/0.15 % OS : Linux 6.8.0-71-generic
% 0.09/0.15 % CPULimit : 300
% 0.09/0.15 % WCLimit : 300
% 0.09/0.15 % DateTime : Tue Oct 6 13:45:36 UTC 2026
% 0.09/0.15 % CPUTime :
% 0.09/0.15 Running leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox/solver/bin/eprover /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.19
% 0.09/0.19 No.of.Axioms: 0
% 0.09/0.19
% 0.09/0.19 Length.of.Defs: 0
% 0.09/0.19
% 0.09/0.19 Contains.Choice.Funs: false
% 0.09/0.19 .
% 0.09/0.19
% 0.09/0.19 ********************************
% 0.09/0.19 * All subproblems solved! *
% 0.09/0.19 ********************************
% 0.09/0.19 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : (rf:0,axioms:5,ps:3,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:true,expand_extuni:false,foatp:e,atp_timeout:15,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:60,loop_count:0,foatp_calls:1,translation:fof_full)
% 0.09/0.19
% 0.09/0.19 %**** Beginning of derivation protocol ****
% 0.09/0.19 % SZS output start CNFRefutation
% See solution above
% 0.09/0.19
% 0.09/0.19 %**** End of derivation protocol ****
% 0.09/0.19 %**** no. of clauses in derivation: 61 ****
% 0.09/0.19 %**** clause counter: 60 ****
% 0.09/0.19
% 0.09/0.19 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : (rf:0,axioms:5,ps:3,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:true,expand_extuni:false,foatp:e,atp_timeout:15,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:60,loop_count:0,foatp_calls:1,translation:fof_full)
%------------------------------------------------------------------------------