%------------------------------------------------------------------------------
% File : LEO-II---2.3.1
% Problem : COL075-1 : TPTP v9.3.1. Released v1.2.0.
% 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 : n016.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 10:11:07 AM UTC 2026
% Result : Unsatisfiable 3.49s 0.93s
% Output : CNFRefutation 3.49s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 12
% Syntax : Number of formulae : 68 ( 63 unt; 0 typ; 0 def)
% Number of atoms : 198 ( 132 equ; 0 cnn)
% Maximal formula atoms : 2 ( 2 avg)
% Number of connectives : 424 ( 18 ~; 14 |; 0 &; 392 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 2 avg)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 13 ( 10 usr; 7 con; 0-2 aty)
% Number of variables : 108 ( 0 ^; 108 !; 0 ?; 108 :)
% Comments :
%------------------------------------------------------------------------------
thf(tp_abstraction,type,
abstraction: $i ).
thf(tp_apply,type,
apply: $i > $i > $i ).
thf(tp_b,type,
b: $i > $i ).
thf(tp_c,type,
c: $i > $i ).
thf(tp_eq,type,
eq: $i ).
thf(tp_k,type,
k: $i ).
thf(tp_n,type,
n: $i > $i > $i ).
thf(tp_pair,type,
pair: $i > $i > $i ).
thf(tp_projection1,type,
projection1: $i ).
thf(tp_projection2,type,
projection2: $i ).
thf(1,axiom,
projection1 != projection2,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',different_projections) ).
thf(2,axiom,
! [X: $i,Y: $i] :
( ( X = Y )
| ( ( apply @ X @ ( n @ X @ Y ) )
!= ( apply @ Y @ ( n @ X @ Y ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',extensionality2) ).
thf(3,axiom,
! [X: $i,Y: $i] :
( ( X = Y )
| ( ( apply @ eq @ ( pair @ X @ Y ) )
= projection2 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',extensionality1) ).
thf(4,axiom,
! [X: $i] :
( ( apply @ eq @ ( pair @ X @ X ) )
= projection1 ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality) ).
thf(5,axiom,
! [X: $i,Y: $i,Z: $i] :
( ( apply @ ( apply @ ( apply @ abstraction @ X ) @ Y ) @ Z )
= ( apply @ ( apply @ X @ ( apply @ k @ Z ) ) @ ( apply @ Y @ Z ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',abstraction) ).
thf(6,axiom,
! [X: $i,Y: $i,Z: $i] :
( ( apply @ ( pair @ X @ Y ) @ Z )
= ( pair @ ( apply @ X @ Z ) @ ( apply @ Y @ Z ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',pairwise_application) ).
thf(7,axiom,
! [X: $i] :
( ( pair @ ( apply @ projection1 @ X ) @ ( apply @ projection2 @ X ) )
= X ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',pairing) ).
thf(8,axiom,
! [X: $i,Y: $i] :
( ( apply @ projection2 @ ( pair @ X @ Y ) )
= Y ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',projection2) ).
thf(9,axiom,
! [X: $i,Y: $i] :
( ( apply @ projection1 @ ( pair @ X @ Y ) )
= X ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',projection1) ).
thf(10,axiom,
! [X: $i,Y: $i] :
( ( apply @ ( apply @ k @ X ) @ Y )
= X ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',k_definition) ).
thf(11,conjecture,
$false,
file('no conjecture given, we try to refute the axioms',dummy_conjecture) ).
thf(12,negated_conjecture,
$false = $false,
inference(negate_conjecture,[status(cth)],[11]) ).
thf(13,negated_conjecture,
! [Y: $i] :
( ( apply @ ( apply @ Y @ ( b @ Y ) ) @ ( c @ Y ) )
!= ( apply @ ( b @ Y ) @ ( b @ Y ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_diagonal_combinator) ).
thf(14,plain,
$false = $false,
inference(unfold_def,[status(thm)],[12]) ).
thf(15,plain,
( ( projection1 != projection2 )
= $true ),
inference(unfold_def,[status(thm)],[1]) ).
thf(16,plain,
( ( ! [X: $i,Y: $i] :
( ( X = Y )
| ( ( apply @ X @ ( n @ X @ Y ) )
!= ( apply @ Y @ ( n @ X @ Y ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[2]) ).
thf(17,plain,
( ( ! [X: $i,Y: $i] :
( ( X = Y )
| ( ( apply @ eq @ ( pair @ X @ Y ) )
= projection2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[3]) ).
thf(18,plain,
( ( ! [X: $i] :
( ( apply @ eq @ ( pair @ X @ X ) )
= projection1 ) )
= $true ),
inference(unfold_def,[status(thm)],[4]) ).
thf(19,plain,
( ( ! [X: $i,Y: $i,Z: $i] :
( ( apply @ ( apply @ ( apply @ abstraction @ X ) @ Y ) @ Z )
= ( apply @ ( apply @ X @ ( apply @ k @ Z ) ) @ ( apply @ Y @ Z ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[5]) ).
thf(20,plain,
( ( ! [X: $i,Y: $i,Z: $i] :
( ( apply @ ( pair @ X @ Y ) @ Z )
= ( pair @ ( apply @ X @ Z ) @ ( apply @ Y @ Z ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[6]) ).
thf(21,plain,
( ( ! [X: $i] :
( ( pair @ ( apply @ projection1 @ X ) @ ( apply @ projection2 @ X ) )
= X ) )
= $true ),
inference(unfold_def,[status(thm)],[7]) ).
thf(22,plain,
( ( ! [X: $i,Y: $i] :
( ( apply @ projection2 @ ( pair @ X @ Y ) )
= Y ) )
= $true ),
inference(unfold_def,[status(thm)],[8]) ).
thf(23,plain,
( ( ! [X: $i,Y: $i] :
( ( apply @ projection1 @ ( pair @ X @ Y ) )
= X ) )
= $true ),
inference(unfold_def,[status(thm)],[9]) ).
thf(24,plain,
( ( ! [X: $i,Y: $i] :
( ( apply @ ( apply @ k @ X ) @ Y )
= X ) )
= $true ),
inference(unfold_def,[status(thm)],[10]) ).
thf(25,plain,
( ( ! [Y: $i] :
( ( apply @ ( apply @ Y @ ( b @ Y ) ) @ ( c @ Y ) )
!= ( apply @ ( b @ Y ) @ ( b @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[13]) ).
thf(26,plain,
( ( ~ $false ) = $true ),
inference(polarity_switch,[status(thm)],[14]) ).
thf(27,plain,
( ( ! [Y: $i] :
( ( apply @ ( apply @ Y @ ( b @ Y ) ) @ ( c @ Y ) )
!= ( apply @ ( b @ Y ) @ ( b @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[25]) ).
thf(28,plain,
( ( ! [X: $i,Y: $i] :
( ( apply @ ( apply @ k @ X ) @ Y )
= X ) )
= $true ),
inference(copy,[status(thm)],[24]) ).
thf(29,plain,
( ( ! [X: $i,Y: $i] :
( ( apply @ projection1 @ ( pair @ X @ Y ) )
= X ) )
= $true ),
inference(copy,[status(thm)],[23]) ).
thf(30,plain,
( ( ! [X: $i,Y: $i] :
( ( apply @ projection2 @ ( pair @ X @ Y ) )
= Y ) )
= $true ),
inference(copy,[status(thm)],[22]) ).
thf(31,plain,
( ( ! [X: $i] :
( ( pair @ ( apply @ projection1 @ X ) @ ( apply @ projection2 @ X ) )
= X ) )
= $true ),
inference(copy,[status(thm)],[21]) ).
thf(32,plain,
( ( ! [X: $i,Y: $i,Z: $i] :
( ( apply @ ( pair @ X @ Y ) @ Z )
= ( pair @ ( apply @ X @ Z ) @ ( apply @ Y @ Z ) ) ) )
= $true ),
inference(copy,[status(thm)],[20]) ).
thf(33,plain,
( ( ! [X: $i,Y: $i,Z: $i] :
( ( apply @ ( apply @ ( apply @ abstraction @ X ) @ Y ) @ Z )
= ( apply @ ( apply @ X @ ( apply @ k @ Z ) ) @ ( apply @ Y @ Z ) ) ) )
= $true ),
inference(copy,[status(thm)],[19]) ).
thf(34,plain,
( ( ! [X: $i] :
( ( apply @ eq @ ( pair @ X @ X ) )
= projection1 ) )
= $true ),
inference(copy,[status(thm)],[18]) ).
thf(35,plain,
( ( ! [X: $i,Y: $i] :
( ( X = Y )
| ( ( apply @ eq @ ( pair @ X @ Y ) )
= projection2 ) ) )
= $true ),
inference(copy,[status(thm)],[17]) ).
thf(36,plain,
( ( ! [X: $i,Y: $i] :
( ( X = Y )
| ( ( apply @ X @ ( n @ X @ Y ) )
!= ( apply @ Y @ ( n @ X @ Y ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[16]) ).
thf(37,plain,
( ( projection1 != projection2 )
= $true ),
inference(copy,[status(thm)],[15]) ).
thf(38,plain,
( ( ~ $false ) = $true ),
inference(copy,[status(thm)],[26]) ).
thf(39,plain,
( ( ! [SX0: $i,SX1: $i] :
( ( SX0 = SX1 )
| ( ( apply @ SX0 @ ( n @ SX0 @ SX1 ) )
!= ( apply @ SX1 @ ( n @ SX0 @ SX1 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[36]) ).
thf(40,plain,
( ( projection1 != projection2 )
= $true ),
inference(unfold_def,[status(thm)],[37]) ).
thf(41,plain,
( ( ! [SX0: $i] :
( ( apply @ ( apply @ SX0 @ ( b @ SX0 ) ) @ ( c @ SX0 ) )
!= ( apply @ ( b @ SX0 ) @ ( b @ SX0 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[27]) ).
thf(42,plain,
! [SV1: $i] :
( ( ! [SY19: $i] :
( ( apply @ ( apply @ k @ SV1 ) @ SY19 )
= SV1 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[28]) ).
thf(43,plain,
! [SV2: $i] :
( ( ! [SY20: $i] :
( ( apply @ projection1 @ ( pair @ SV2 @ SY20 ) )
= SV2 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[29]) ).
thf(44,plain,
! [SV3: $i] :
( ( ! [SY21: $i] :
( ( apply @ projection2 @ ( pair @ SV3 @ SY21 ) )
= SY21 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[30]) ).
thf(45,plain,
! [SV4: $i] :
( ( ( pair @ ( apply @ projection1 @ SV4 ) @ ( apply @ projection2 @ SV4 ) )
= SV4 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[31]) ).
thf(46,plain,
! [SV5: $i] :
( ( ! [SY22: $i,SY23: $i] :
( ( apply @ ( pair @ SV5 @ SY22 ) @ SY23 )
= ( pair @ ( apply @ SV5 @ SY23 ) @ ( apply @ SY22 @ SY23 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[32]) ).
thf(47,plain,
! [SV6: $i] :
( ( ! [SY24: $i,SY25: $i] :
( ( apply @ ( apply @ ( apply @ abstraction @ SV6 ) @ SY24 ) @ SY25 )
= ( apply @ ( apply @ SV6 @ ( apply @ k @ SY25 ) ) @ ( apply @ SY24 @ SY25 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[33]) ).
thf(48,plain,
! [SV7: $i] :
( ( ( apply @ eq @ ( pair @ SV7 @ SV7 ) )
= projection1 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[34]) ).
thf(49,plain,
! [SV8: $i] :
( ( ! [SY26: $i] :
( ( SV8 = SY26 )
| ( ( apply @ eq @ ( pair @ SV8 @ SY26 ) )
= projection2 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[35]) ).
thf(50,plain,
$false = $false,
inference(extcnf_not_pos,[status(thm)],[38]) ).
thf(51,plain,
! [SV9: $i] :
( ( ! [SY27: $i] :
( ( SV9 = SY27 )
| ( ( apply @ SV9 @ ( n @ SV9 @ SY27 ) )
!= ( apply @ SY27 @ ( n @ SV9 @ SY27 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[39]) ).
thf(52,plain,
( ( projection1 = projection2 )
= $false ),
inference(extcnf_not_pos,[status(thm)],[40]) ).
thf(53,plain,
! [SV10: $i] :
( ( ( apply @ ( apply @ SV10 @ ( b @ SV10 ) ) @ ( c @ SV10 ) )
!= ( apply @ ( b @ SV10 ) @ ( b @ SV10 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[41]) ).
thf(54,plain,
! [SV11: $i,SV1: $i] :
( ( ( apply @ ( apply @ k @ SV1 ) @ SV11 )
= SV1 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[42]) ).
thf(55,plain,
! [SV12: $i,SV2: $i] :
( ( ( apply @ projection1 @ ( pair @ SV2 @ SV12 ) )
= SV2 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[43]) ).
thf(56,plain,
! [SV13: $i,SV3: $i] :
( ( ( apply @ projection2 @ ( pair @ SV3 @ SV13 ) )
= SV13 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[44]) ).
thf(57,plain,
! [SV14: $i,SV5: $i] :
( ( ! [SY28: $i] :
( ( apply @ ( pair @ SV5 @ SV14 ) @ SY28 )
= ( pair @ ( apply @ SV5 @ SY28 ) @ ( apply @ SV14 @ SY28 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[46]) ).
thf(58,plain,
! [SV15: $i,SV6: $i] :
( ( ! [SY29: $i] :
( ( apply @ ( apply @ ( apply @ abstraction @ SV6 ) @ SV15 ) @ SY29 )
= ( apply @ ( apply @ SV6 @ ( apply @ k @ SY29 ) ) @ ( apply @ SV15 @ SY29 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[47]) ).
thf(59,plain,
! [SV16: $i,SV8: $i] :
( ( ( SV8 = SV16 )
| ( ( apply @ eq @ ( pair @ SV8 @ SV16 ) )
= projection2 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[49]) ).
thf(60,plain,
! [SV17: $i,SV9: $i] :
( ( ( SV9 = SV17 )
| ( ( apply @ SV9 @ ( n @ SV9 @ SV17 ) )
!= ( apply @ SV17 @ ( n @ SV9 @ SV17 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[51]) ).
thf(61,plain,
! [SV10: $i] :
( ( ( apply @ ( apply @ SV10 @ ( b @ SV10 ) ) @ ( c @ SV10 ) )
= ( apply @ ( b @ SV10 ) @ ( b @ SV10 ) ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[53]) ).
thf(62,plain,
! [SV18: $i,SV14: $i,SV5: $i] :
( ( ( apply @ ( pair @ SV5 @ SV14 ) @ SV18 )
= ( pair @ ( apply @ SV5 @ SV18 ) @ ( apply @ SV14 @ SV18 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[57]) ).
thf(63,plain,
! [SV19: $i,SV15: $i,SV6: $i] :
( ( ( apply @ ( apply @ ( apply @ abstraction @ SV6 ) @ SV15 ) @ SV19 )
= ( apply @ ( apply @ SV6 @ ( apply @ k @ SV19 ) ) @ ( apply @ SV15 @ SV19 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[58]) ).
thf(64,plain,
! [SV16: $i,SV8: $i] :
( ( ( SV8 = SV16 )
= $true )
| ( ( ( apply @ eq @ ( pair @ SV8 @ SV16 ) )
= projection2 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[59]) ).
thf(65,plain,
! [SV17: $i,SV9: $i] :
( ( ( SV9 = SV17 )
= $true )
| ( ( ( apply @ SV9 @ ( n @ SV9 @ SV17 ) )
!= ( apply @ SV17 @ ( n @ SV9 @ SV17 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[60]) ).
thf(66,plain,
! [SV17: $i,SV9: $i] :
( ( ( ( apply @ SV9 @ ( n @ SV9 @ SV17 ) )
= ( apply @ SV17 @ ( n @ SV9 @ SV17 ) ) )
= $false )
| ( ( SV9 = SV17 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[65]) ).
thf(67,plain,
$false = $true,
inference(fo_atp_e,[status(thm)],[45,66,64,63,62,61,56,55,54,52,50,48]) ).
thf(68,plain,
$false,
inference(solved_all_splits,[solved_all_splits(join,[])],[67]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : COL075-1 : TPTP v9.3.1. Released v1.2.0.
% 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
% 0.09/0.14 % Computer : n016.cluster.edu
% 0.09/0.14 % Model : x86_64 x86_64
% 0.09/0.14 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.14 % Memory : 8046.5625MB
% 0.09/0.14 % OS : Linux 6.8.0-71-generic
% 0.09/0.14 % CPULimit : 300
% 0.09/0.14 % WCLimit : 300
% 0.09/0.14 % DateTime : Tue Oct 6 18:39:29 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
% 3.49/0.93
% 3.49/0.93 No.of.Axioms: 11
% 3.49/0.93
% 3.49/0.93 Length.of.Defs: 0
% 3.49/0.93
% 3.49/0.93 Contains.Choice.Funs: false
% 3.49/0.93 .
% 3.49/0.93
% 3.49/0.93 ********************************
% 3.49/0.93 * All subproblems solved! *
% 3.49/0.93 ********************************
% 3.49/0.93 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : (rf:1,axioms:11,ps:0,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:false,expand_extuni:false,foatp:e,atp_timeout:299,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:67,loop_count:0,foatp_calls:1,translation:fof_full)
% 3.49/0.93
% 3.49/0.93 %**** Beginning of derivation protocol ****
% 3.49/0.93 % SZS output start CNFRefutation
% See solution above
% 3.49/0.93
% 3.49/0.93 %**** End of derivation protocol ****
% 3.49/0.93 %**** no. of clauses in derivation: 68 ****
% 3.49/0.93 %**** clause counter: 67 ****
% 3.49/0.93
% 3.49/0.93 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : (rf:1,axioms:11,ps:0,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:false,expand_extuni:false,foatp:e,atp_timeout:299,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:67,loop_count:0,foatp_calls:1,translation:fof_full)
%------------------------------------------------------------------------------