%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : PUZ062-2 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 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 : Sun Sep 27 08:23:15 AM UTC 2026
% Result : Unsatisfiable 0.09s 10.79s
% Output : CNFRefutation 0.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 13
% Syntax : Number of formulae : 53 ( 40 unt; 0 def)
% Number of atoms : 70 ( 25 equ)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 44 ( 27 ~; 17 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 2 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-3 aty)
% Number of functors : 20 ( 20 usr; 11 con; 0-4 aty)
% Number of variables : 51 ( 10 sgn 20 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
fof(cls_Finite__Set_Ocard__insert__disjoint_0,axiom,
! [VAR_0,VAR_1,VAR_2] :
( c_Finite__Set_Ocard(c_insert(VAR_2,VAR_0,VAR_1),VAR_1) = c_Suc(c_Finite__Set_Ocard(VAR_0,VAR_1))
| c_in(VAR_2,VAR_0,VAR_1)
| ~ c_in(VAR_0,c_Finite__Set_OFinites,tc_set(VAR_1)) ),
file('theBenchmark.p',cls_Finite__Set_Ocard__insert__disjoint_0) ).
cnf(c_1,plain,
( c_Finite__Set_Ocard(c_insert(C,A,B),B) = c_Suc(c_Finite__Set_Ocard(A,B))
| c_in(C,A,B)
| ~ c_in(A,c_Finite__Set_OFinites,tc_set(B)) ),
inference(clausify,[status(thm)],[cls_Finite__Set_Ocard__insert__disjoint_0]) ).
fof(cls_Finite__Set_Ofinite__Int_1,axiom,
! [VAR_0,VAR_1,VAR_2] :
( c_in(c_inter(VAR_2,VAR_0,VAR_1),c_Finite__Set_OFinites,tc_set(VAR_1))
| ~ c_in(VAR_0,c_Finite__Set_OFinites,tc_set(VAR_1)) ),
file('theBenchmark.p',cls_Finite__Set_Ofinite__Int_1) ).
cnf(c_2,plain,
( c_in(c_inter(C,A,B),c_Finite__Set_OFinites,tc_set(B))
| ~ c_in(A,c_Finite__Set_OFinites,tc_set(B)) ),
inference(clausify,[status(thm)],[cls_Finite__Set_Ofinite__Int_1]) ).
fof(cls_Mutil_Otiling__domino__finite_0,axiom,
! [VAR_0] :
( c_in(VAR_0,c_Finite__Set_OFinites,tc_set(tc_prod(tc_nat,tc_nat)))
| ~ c_in(VAR_0,c_Mutil_Otiling(c_Mutil_Odomino,tc_prod(tc_nat,tc_nat)),tc_set(tc_prod(tc_nat,tc_nat))) ),
file('theBenchmark.p',cls_Mutil_Otiling__domino__finite_0) ).
cnf(c_3,plain,
( c_in(A,c_Finite__Set_OFinites,tc_set(tc_prod(tc_nat,tc_nat)))
| ~ c_in(A,c_Mutil_Otiling(c_Mutil_Odomino,tc_prod(tc_nat,tc_nat)),tc_set(tc_prod(tc_nat,tc_nat))) ),
inference(clausify,[status(thm)],[cls_Mutil_Otiling__domino__finite_0]) ).
fof(cls_Set_OInt__iff_1,axiom,
! [VAR_0,VAR_1,VAR_2,VAR_3] :
( c_in(VAR_0,VAR_2,VAR_3)
| ~ c_in(VAR_0,c_inter(VAR_1,VAR_2,VAR_3),VAR_3) ),
file('theBenchmark.p',cls_Set_OInt__iff_1) ).
cnf(c_4,plain,
( c_in(A,C,D)
| ~ c_in(A,c_inter(B,C,D),D) ),
inference(clausify,[status(thm)],[cls_Set_OInt__iff_1]) ).
fof(cls_Set_OInt__iff_2,axiom,
! [VAR_0,VAR_1,VAR_2,VAR_3] :
( c_in(VAR_0,c_inter(VAR_3,VAR_1,VAR_2),VAR_2)
| ~ c_in(VAR_0,VAR_3,VAR_2)
| ~ c_in(VAR_0,VAR_1,VAR_2) ),
file('theBenchmark.p',cls_Set_OInt__iff_2) ).
cnf(c_5,plain,
( c_in(A,c_inter(D,B,C),C)
| ~ c_in(A,D,C)
| ~ c_in(A,B,C) ),
inference(clausify,[status(thm)],[cls_Set_OInt__iff_2]) ).
fof(cls_Set_Oinsert__iff_1,axiom,
! [VAR_0,VAR_1,VAR_2] : c_in(VAR_0,c_insert(VAR_0,VAR_1,VAR_2),VAR_2),
file('theBenchmark.p',cls_Set_Oinsert__iff_1) ).
cnf(c_6,plain,
c_in(A,c_insert(A,B,C),C),
inference(clausify,[status(thm)],[cls_Set_Oinsert__iff_1]) ).
fof(cls_Set_Oempty__iff_0,axiom,
! [VAR_0,VAR_1] : ~ c_in(VAR_0,c_emptyset,VAR_1),
file('theBenchmark.p',cls_Set_Oempty__iff_0) ).
cnf(c_7,plain,
~ c_in(A,c_emptyset,B),
inference(clausify,[status(thm)],[cls_Set_Oempty__iff_0]) ).
fof(cls_conjecture_0,axiom,
c_in(v_t,c_Mutil_Otiling(c_Mutil_Odomino,tc_prod(tc_nat,tc_nat)),tc_set(tc_prod(tc_nat,tc_nat))),
file('theBenchmark.p',cls_conjecture_0) ).
cnf(c_8,plain,
c_in(v_t,c_Mutil_Otiling(c_Mutil_Odomino,tc_prod(tc_nat,tc_nat)),tc_set(tc_prod(tc_nat,tc_nat))),
inference(clausify,[status(thm)],[cls_conjecture_0]) ).
fof(cls_conjecture_1,axiom,
c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) = c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
file('theBenchmark.p',cls_conjecture_1) ).
cnf(c_9,plain,
c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) = c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(clausify,[status(thm)],[cls_conjecture_1]) ).
cnf(c_10,plain,
c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) = c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(copy,[status(thm)],[c_9]) ).
fof(cls_conjecture_2,axiom,
c_inter(v_a,v_t,tc_prod(tc_nat,tc_nat)) = c_emptyset,
file('theBenchmark.p',cls_conjecture_2) ).
cnf(c_11,plain,
c_inter(v_a,v_t,tc_prod(tc_nat,tc_nat)) = c_emptyset,
inference(clausify,[status(thm)],[cls_conjecture_2]) ).
cnf(c_12,plain,
c_emptyset = c_inter(v_a,v_t,tc_prod(tc_nat,tc_nat)),
inference(copy,[status(thm)],[c_11]) ).
fof(cls_conjecture_3,axiom,
c_inter(c_Mutil_Ocoloured(c_0),v_a,tc_prod(tc_nat,tc_nat)) = c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_emptyset,tc_prod(tc_nat,tc_nat)),
file('theBenchmark.p',cls_conjecture_3) ).
cnf(c_13,plain,
c_inter(c_Mutil_Ocoloured(c_0),v_a,tc_prod(tc_nat,tc_nat)) = c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_emptyset,tc_prod(tc_nat,tc_nat)),
inference(clausify,[status(thm)],[cls_conjecture_3]) ).
cnf(c_460,plain,
c_inter(c_Mutil_Ocoloured(c_0),v_a,tc_prod(tc_nat,tc_nat)) = c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(v_a,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(paramod,[status(thm)],[c_12,c_13]) ).
cnf(c_14,plain,
c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(v_a,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) = c_inter(c_Mutil_Ocoloured(c_0),v_a,tc_prod(tc_nat,tc_nat)),
inference(copy,[status(thm)],[c_460]) ).
fof(cls_conjecture_4,axiom,
c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_a,tc_prod(tc_nat,tc_nat)) = c_insert(c_Pair(v_m,v_n,tc_nat,tc_nat),c_emptyset,tc_prod(tc_nat,tc_nat)),
file('theBenchmark.p',cls_conjecture_4) ).
cnf(c_15,plain,
c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_a,tc_prod(tc_nat,tc_nat)) = c_insert(c_Pair(v_m,v_n,tc_nat,tc_nat),c_emptyset,tc_prod(tc_nat,tc_nat)),
inference(clausify,[status(thm)],[cls_conjecture_4]) ).
cnf(c_461,plain,
c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_a,tc_prod(tc_nat,tc_nat)) = c_insert(c_Pair(v_m,v_n,tc_nat,tc_nat),c_inter(v_a,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(paramod,[status(thm)],[c_12,c_15]) ).
cnf(c_16,plain,
c_insert(c_Pair(v_m,v_n,tc_nat,tc_nat),c_inter(v_a,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) = c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_a,tc_prod(tc_nat,tc_nat)),
inference(copy,[status(thm)],[c_461]) ).
fof(cls_conjecture_5,axiom,
c_Finite__Set_Ocard(c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) != c_Finite__Set_Ocard(c_insert(c_Pair(v_m,v_n,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
file('theBenchmark.p',cls_conjecture_5) ).
cnf(c_17,plain,
c_Finite__Set_Ocard(c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) != c_Finite__Set_Ocard(c_insert(c_Pair(v_m,v_n,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(clausify,[status(thm)],[cls_conjecture_5]) ).
cnf(c_18,plain,
c_Finite__Set_Ocard(c_insert(c_Pair(v_m,v_n,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) != c_Finite__Set_Ocard(c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(copy,[status(thm)],[c_17]) ).
cnf(c_20,plain,
~ c_in(A,c_inter(v_a,v_t,tc_prod(tc_nat,tc_nat)),B),
inference(paramod,[status(thm)],[c_12,c_7]) ).
cnf(c_25,plain,
c_in(v_t,c_Finite__Set_OFinites,tc_set(tc_prod(tc_nat,tc_nat))),
inference(resolve,[status(thm)],[c_8,c_3]) ).
cnf(c_26,plain,
c_in(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_0),v_a,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(paramod,[status(thm)],[c_14,c_6]) ).
cnf(c_27,plain,
c_in(c_Pair(v_m,v_n,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_a,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(paramod,[status(thm)],[c_16,c_6]) ).
cnf(c_41,plain,
c_in(c_inter(A,v_t,tc_prod(tc_nat,tc_nat)),c_Finite__Set_OFinites,tc_set(tc_prod(tc_nat,tc_nat))),
inference(resolve,[status(thm)],[c_25,c_2]) ).
cnf(c_48,plain,
( c_Finite__Set_Ocard(c_insert(A,c_inter(B,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) = c_Suc(c_Finite__Set_Ocard(c_inter(B,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)))
| c_in(A,c_inter(B,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) ),
inference(resolve,[status(thm)],[c_41,c_1]) ).
cnf(c_73,plain,
c_in(c_Pair(v_i,v_j,tc_nat,tc_nat),v_a,tc_prod(tc_nat,tc_nat)),
inference(resolve,[status(thm)],[c_26,c_4]) ).
cnf(c_462,plain,
( c_in(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(v_a,A,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat))
| ~ c_in(c_Pair(v_i,v_j,tc_nat,tc_nat),A,tc_prod(tc_nat,tc_nat)) ),
inference(resolve,[status(thm)],[c_5,c_73]) ).
cnf(c_79,plain,
~ c_in(c_Pair(v_i,v_j,tc_nat,tc_nat),v_t,tc_prod(tc_nat,tc_nat)),
inference(resolve,[status(thm)],[c_462,c_20]) ).
cnf(c_88,plain,
c_in(c_Pair(v_m,v_n,tc_nat,tc_nat),v_a,tc_prod(tc_nat,tc_nat)),
inference(resolve,[status(thm)],[c_27,c_4]) ).
cnf(c_92,plain,
~ c_in(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(A,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(resolve,[status(thm)],[c_4,c_79]) ).
cnf(c_463,plain,
( c_in(c_Pair(v_m,v_n,tc_nat,tc_nat),c_inter(v_a,A,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat))
| ~ c_in(c_Pair(v_m,v_n,tc_nat,tc_nat),A,tc_prod(tc_nat,tc_nat)) ),
inference(resolve,[status(thm)],[c_5,c_88]) ).
cnf(c_101,plain,
~ c_in(c_Pair(v_m,v_n,tc_nat,tc_nat),v_t,tc_prod(tc_nat,tc_nat)),
inference(resolve,[status(thm)],[c_463,c_20]) ).
cnf(c_124,plain,
~ c_in(c_Pair(v_m,v_n,tc_nat,tc_nat),c_inter(A,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(resolve,[status(thm)],[c_4,c_101]) ).
cnf(c_448,plain,
c_Finite__Set_Ocard(c_insert(c_Pair(v_m,v_n,tc_nat,tc_nat),c_inter(A,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) = c_Suc(c_Finite__Set_Ocard(c_inter(A,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat))),
inference(resolve,[status(thm)],[c_48,c_124]) ).
cnf(c_449,plain,
c_Finite__Set_Ocard(c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(A,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)) = c_Suc(c_Finite__Set_Ocard(c_inter(A,v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat))),
inference(resolve,[status(thm)],[c_48,c_92]) ).
cnf(c_464,plain,
c_Suc(c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_Suc(c_0)),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat))) != c_Finite__Set_Ocard(c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(paramod,[status(thm)],[c_448,c_18]) ).
cnf(c_465,plain,
c_Suc(c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat))) != c_Finite__Set_Ocard(c_insert(c_Pair(v_i,v_j,tc_nat,tc_nat),c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat)),
inference(paramod,[status(thm)],[c_10,c_464]) ).
cnf(c_466,plain,
c_Suc(c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat))) != c_Suc(c_Finite__Set_Ocard(c_inter(c_Mutil_Ocoloured(c_0),v_t,tc_prod(tc_nat,tc_nat)),tc_prod(tc_nat,tc_nat))),
inference(paramod,[status(thm)],[c_449,c_465]) ).
cnf(c_459,plain,
$false,
inference(copy,[status(thm)],[c_466]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.01 % Problem : PUZ062-2 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.02 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.07/10.58 % Computer : n008.cluster.edu
% 0.07/10.58 % Model : x86_64 x86_64
% 0.07/10.58 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/10.58 % Memory : 8046.5625MB
% 0.07/10.58 % OS : Linux 6.8.0-71-generic
% 0.07/10.58 % CPULimit : 300
% 0.07/10.58 % WCLimit : 300
% 0.07/10.58 % DateTime : Sat Sep 26 05:11:37 UTC 2026
% 0.07/10.58 % CPUTime :
% 0.07/10.58 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.07/10.58 % Process 2837198 was started by sandbox on n008,
% 0.07/10.58 % Sat Sep 26 05:11:37 2026
% 0.07/10.58 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.07/10.59
% 0.07/10.59 % From the command line: assign(max_seconds, 300).
% 0.09/10.79
% 0.09/10.79 % SZS status Unsatisfiable for theBenchmark
% 0.09/10.79
% 0.09/10.79 % Proof 1 at 0.05 (+ 0.01) seconds.
% 0.09/10.79 % Length of proof is 33.
% 0.09/10.79 % Level of proof is 8.
% 0.09/10.79 % Maximum clause weight is 42.000.
% 0.09/10.79 % Given clauses 105.
% 0.09/10.79
% 0.09/10.79 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------