%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : GRP774+1 : TPTP v9.3.1. Released v4.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n010.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 : Tue Sep 29 10:11:58 AM UTC 2026
% Result : Theorem 0.18s 0.52s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : GRP774+1 : TPTP v9.3.1. Released v4.1.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.37 % Computer : n010.cluster.edu
% 0.09/0.37 % Model : x86_64 x86_64
% 0.09/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.37 % Memory : 8046.5625MB
% 0.09/0.37 % OS : Linux 6.8.0-71-generic
% 0.09/0.37 % CPULimit : 300
% 0.09/0.37 % WCLimit : 300
% 0.09/0.37 % DateTime : Sun Sep 27 10:44:16 UTC 2026
% 0.09/0.38 % CPUTime :
% 0.09/0.38 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.18/0.52 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.18/0.52
% 0.18/0.52 % SZS status Theorem
% 0.18/0.52
% 0.18/0.54 % SZS output start Proof
% 0.18/0.54 Axiom 1 (sos02): product(X, X) = X.
% 0.18/0.54 Axiom 2 (goals): d(x4, x5) = true.
% 0.18/0.54 Axiom 3 (goals_1): d(x2, x3) = true.
% 0.18/0.54 Axiom 4 (sos01): product(product(X, Y), Z) = product(X, product(Y, Z)).
% 0.18/0.54 Axiom 5 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.18/0.54 Axiom 6 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 0.18/0.54 Axiom 7 (sos03_1): ifeq(d(X, Y), true, product(X, product(Y, X)), X) = X.
% 0.18/0.54 Axiom 8 (sos03_2): ifeq(d(X, Y), true, product(Y, product(X, Y)), Y) = Y.
% 0.18/0.54 Axiom 9 (sos03): ifeq2(product(X, product(Y, X)), X, ifeq2(product(Y, product(X, Y)), Y, d(Y, X), true), true) = true.
% 0.18/0.54
% 0.18/0.54 Lemma 10: product(X, product(Y, product(X, Y))) = product(X, Y).
% 0.18/0.54 Proof:
% 0.18/0.54 product(X, product(Y, product(X, Y)))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(X, Y), product(X, Y))
% 0.18/0.54 = { by axiom 1 (sos02) }
% 0.18/0.54 product(X, Y)
% 0.18/0.54
% 0.18/0.54 Lemma 11: product(x4, product(x5, product(x4, X))) = product(x4, X).
% 0.18/0.54 Proof:
% 0.18/0.54 product(x4, product(x5, product(x4, X)))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(x4, product(product(x5, x4), X))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(x4, product(x5, x4)), X)
% 0.18/0.54 = { by axiom 5 (ifeq_axiom) R->L }
% 0.18/0.54 product(ifeq(true, true, product(x4, product(x5, x4)), x4), X)
% 0.18/0.54 = { by axiom 2 (goals) R->L }
% 0.18/0.54 product(ifeq(d(x4, x5), true, product(x4, product(x5, x4)), x4), X)
% 0.18/0.54 = { by axiom 7 (sos03_1) }
% 0.18/0.54 product(x4, X)
% 0.18/0.54
% 0.18/0.54 Lemma 12: product(x5, product(x4, product(x5, X))) = product(x5, X).
% 0.18/0.54 Proof:
% 0.18/0.54 product(x5, product(x4, product(x5, X)))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(x5, product(product(x4, x5), X))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(x5, product(x4, x5)), X)
% 0.18/0.54 = { by axiom 5 (ifeq_axiom) R->L }
% 0.18/0.54 product(ifeq(true, true, product(x5, product(x4, x5)), x5), X)
% 0.18/0.54 = { by axiom 2 (goals) R->L }
% 0.18/0.54 product(ifeq(d(x4, x5), true, product(x5, product(x4, x5)), x5), X)
% 0.18/0.54 = { by axiom 8 (sos03_2) }
% 0.18/0.54 product(x5, X)
% 0.18/0.54
% 0.18/0.54 Lemma 13: product(x2, product(x3, product(x2, X))) = product(x2, X).
% 0.18/0.54 Proof:
% 0.18/0.54 product(x2, product(x3, product(x2, X)))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(x2, product(product(x3, x2), X))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(x2, product(x3, x2)), X)
% 0.18/0.54 = { by axiom 5 (ifeq_axiom) R->L }
% 0.18/0.54 product(ifeq(true, true, product(x2, product(x3, x2)), x2), X)
% 0.18/0.54 = { by axiom 3 (goals_1) R->L }
% 0.18/0.54 product(ifeq(d(x2, x3), true, product(x2, product(x3, x2)), x2), X)
% 0.18/0.54 = { by axiom 7 (sos03_1) }
% 0.18/0.54 product(x2, X)
% 0.18/0.54
% 0.18/0.54 Lemma 14: product(x3, product(x2, product(x3, X))) = product(x3, X).
% 0.18/0.54 Proof:
% 0.18/0.54 product(x3, product(x2, product(x3, X)))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(x3, product(product(x2, x3), X))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(x3, product(x2, x3)), X)
% 0.18/0.54 = { by axiom 5 (ifeq_axiom) R->L }
% 0.18/0.54 product(ifeq(true, true, product(x3, product(x2, x3)), x3), X)
% 0.18/0.54 = { by axiom 3 (goals_1) R->L }
% 0.18/0.54 product(ifeq(d(x2, x3), true, product(x3, product(x2, x3)), x3), X)
% 0.18/0.54 = { by axiom 8 (sos03_2) }
% 0.18/0.54 product(x3, X)
% 0.18/0.54
% 0.18/0.54 Lemma 15: product(product(x2, X), product(x3, product(x2, X))) = product(x2, X).
% 0.18/0.54 Proof:
% 0.18/0.54 product(product(x2, X), product(x3, product(x2, X)))
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(x2, product(X, product(x3, product(x2, X))))
% 0.18/0.54 = { by lemma 13 R->L }
% 0.18/0.54 product(x2, product(x3, product(x2, product(X, product(x3, product(x2, X))))))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(x2, product(x3, product(product(x2, X), product(x3, product(x2, X)))))
% 0.18/0.54 = { by lemma 10 }
% 0.18/0.54 product(x2, product(x3, product(x2, X)))
% 0.18/0.54 = { by lemma 13 }
% 0.18/0.54 product(x2, X)
% 0.18/0.54
% 0.18/0.54 Lemma 16: product(product(x3, X), product(x2, product(x3, X))) = product(x3, X).
% 0.18/0.54 Proof:
% 0.18/0.54 product(product(x3, X), product(x2, product(x3, X)))
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(x3, product(X, product(x2, product(x3, X))))
% 0.18/0.54 = { by lemma 14 R->L }
% 0.18/0.54 product(x3, product(x2, product(x3, product(X, product(x2, product(x3, X))))))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(x3, product(x2, product(product(x3, X), product(x2, product(x3, X)))))
% 0.18/0.54 = { by lemma 10 }
% 0.18/0.54 product(x3, product(x2, product(x3, X)))
% 0.18/0.54 = { by lemma 14 }
% 0.18/0.54 product(x3, X)
% 0.18/0.54
% 0.18/0.54 Lemma 17: product(X, product(Y, product(Z, product(X, product(Y, Z))))) = product(X, product(Y, Z)).
% 0.18/0.54 Proof:
% 0.18/0.54 product(X, product(Y, product(Z, product(X, product(Y, Z)))))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(X, product(Y, product(Z, product(product(X, Y), Z))))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(X, Y), product(Z, product(product(X, Y), Z)))
% 0.18/0.54 = { by lemma 10 }
% 0.18/0.54 product(product(X, Y), Z)
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(X, product(Y, Z))
% 0.18/0.54
% 0.18/0.54 Lemma 18: product(product(X, Y), product(Z, product(Y, product(Z, W)))) = product(product(X, Y), product(Z, W)).
% 0.18/0.54 Proof:
% 0.18/0.54 product(product(X, Y), product(Z, product(Y, product(Z, W))))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(X, Y), product(Z, product(product(Y, Z), W)))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(X, Y), product(product(Z, product(Y, Z)), W))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(product(X, Y), product(Z, product(Y, Z))), W)
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(product(X, product(Y, product(Z, product(Y, Z)))), W)
% 0.18/0.54 = { by lemma 10 }
% 0.18/0.54 product(product(X, product(Y, Z)), W)
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(product(X, Y), Z), W)
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(product(X, Y), product(Z, W))
% 0.18/0.54
% 0.18/0.54 Lemma 19: product(product(X, x4), product(product(Y, x5), product(x4, Y))) = product(product(X, x4), Y).
% 0.18/0.54 Proof:
% 0.18/0.54 product(product(X, x4), product(product(Y, x5), product(x4, Y)))
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(X, product(x4, product(product(Y, x5), product(x4, Y))))
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(X, product(x4, product(Y, product(x5, product(x4, Y)))))
% 0.18/0.54 = { by lemma 11 R->L }
% 0.18/0.54 product(X, product(x4, product(x5, product(x4, product(Y, product(x5, product(x4, Y)))))))
% 0.18/0.54 = { by lemma 17 }
% 0.18/0.54 product(X, product(x4, product(x5, product(x4, Y))))
% 0.18/0.54 = { by lemma 11 }
% 0.18/0.54 product(X, product(x4, Y))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(X, x4), Y)
% 0.18/0.54
% 0.18/0.54 Lemma 20: product(product(x2, X), product(x3, product(product(x2, X), Y))) = product(product(x2, X), Y).
% 0.18/0.54 Proof:
% 0.18/0.54 product(product(x2, X), product(x3, product(product(x2, X), Y)))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(x2, X), product(product(x3, product(x2, X)), Y))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(product(x2, X), product(x3, product(x2, X))), Y)
% 0.18/0.54 = { by lemma 15 }
% 0.18/0.54 product(product(x2, X), Y)
% 0.18/0.54
% 0.18/0.54 Lemma 21: product(product(X, x5), product(product(Y, x4), product(x5, product(Y, Z)))) = product(product(X, x5), product(Y, Z)).
% 0.18/0.54 Proof:
% 0.18/0.54 product(product(X, x5), product(product(Y, x4), product(x5, product(Y, Z))))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(X, x5), product(product(Y, x4), product(product(x5, Y), Z)))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(X, x5), product(product(product(Y, x4), product(x5, Y)), Z))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(product(X, x5), product(product(Y, x4), product(x5, Y))), Z)
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(product(X, product(x5, product(product(Y, x4), product(x5, Y)))), Z)
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(product(X, product(x5, product(Y, product(x4, product(x5, Y))))), Z)
% 0.18/0.54 = { by lemma 12 R->L }
% 0.18/0.54 product(product(X, product(x5, product(x4, product(x5, product(Y, product(x4, product(x5, Y))))))), Z)
% 0.18/0.54 = { by lemma 17 }
% 0.18/0.54 product(product(X, product(x5, product(x4, product(x5, Y)))), Z)
% 0.18/0.54 = { by lemma 12 }
% 0.18/0.54 product(product(X, product(x5, Y)), Z)
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(product(X, x5), Y), Z)
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(product(X, x5), product(Y, Z))
% 0.18/0.54
% 0.18/0.54 Lemma 22: product(product(X, x4), product(product(Y, x5), product(product(X, x4), product(Y, Z)))) = product(product(X, x4), product(Y, Z)).
% 0.18/0.54 Proof:
% 0.18/0.54 product(product(X, x4), product(product(Y, x5), product(product(X, x4), product(Y, Z))))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(X, x4), product(product(Y, x5), product(product(product(X, x4), Y), Z)))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(X, x4), product(product(product(Y, x5), product(product(X, x4), Y)), Z))
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(product(X, x4), product(product(Y, x5), product(product(X, x4), Y))), Z)
% 0.18/0.54 = { by lemma 19 R->L }
% 0.18/0.54 product(product(product(X, x4), product(product(Y, x5), product(product(X, x4), product(product(Y, x5), product(x4, Y))))), Z)
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(product(X, x4), product(product(Y, x5), product(product(product(X, x4), product(Y, x5)), product(x4, Y)))), Z)
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(product(product(X, x4), product(Y, x5)), product(product(product(X, x4), product(Y, x5)), product(x4, Y))), Z)
% 0.18/0.54 = { by axiom 4 (sos01) R->L }
% 0.18/0.54 product(product(product(product(product(X, x4), product(Y, x5)), product(product(X, x4), product(Y, x5))), product(x4, Y)), Z)
% 0.18/0.54 = { by axiom 1 (sos02) }
% 0.18/0.54 product(product(product(product(X, x4), product(Y, x5)), product(x4, Y)), Z)
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(product(product(X, x4), product(product(Y, x5), product(x4, Y))), Z)
% 0.18/0.54 = { by lemma 19 }
% 0.18/0.54 product(product(product(X, x4), Y), Z)
% 0.18/0.54 = { by axiom 4 (sos01) }
% 0.18/0.54 product(product(X, x4), product(Y, Z))
% 0.18/0.54
% 0.18/0.54 Goal 1 (goals_2): d(product(x2, x4), product(x3, x5)) = true.
% 0.18/0.54 Proof:
% 0.18/0.54 d(product(x2, x4), product(x3, x5))
% 0.18/0.54 = { by axiom 6 (ifeq_axiom) R->L }
% 0.18/0.54 ifeq2(product(x2, x4), product(x2, x4), d(product(x2, x4), product(x3, x5)), true)
% 0.18/0.54 = { by lemma 15 R->L }
% 0.18/0.54 ifeq2(product(product(x2, x4), product(x3, product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true)
% 0.18/0.54 = { by lemma 22 R->L }
% 0.18/0.54 ifeq2(product(product(x2, x4), product(product(x3, x5), product(product(x2, x4), product(x3, product(x2, x4))))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true)
% 0.18/0.54 = { by lemma 15 }
% 0.18/0.54 ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true)
% 0.18/0.54 = { by axiom 6 (ifeq_axiom) R->L }
% 0.18/0.54 ifeq2(product(x3, x5), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.54 = { by lemma 16 R->L }
% 0.18/0.54 ifeq2(product(product(x3, x5), product(x2, product(x3, x5))), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.55 = { by lemma 21 R->L }
% 0.18/0.55 ifeq2(product(product(x3, x5), product(product(x2, x4), product(x5, product(x2, product(x3, x5))))), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.55 = { by lemma 18 R->L }
% 0.18/0.55 ifeq2(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(x5, product(x2, product(x3, x5))))))), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.55 = { by lemma 20 R->L }
% 0.18/0.55 ifeq2(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(x3, product(product(x2, x4), product(x5, product(x2, product(x3, x5))))))))), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.55 = { by lemma 22 R->L }
% 0.18/0.55 ifeq2(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(product(x3, x5), product(product(x2, x4), product(x3, product(product(x2, x4), product(x5, product(x2, product(x3, x5))))))))))), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.55 = { by lemma 20 }
% 0.18/0.55 ifeq2(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(product(x3, x5), product(product(x2, x4), product(x5, product(x2, product(x3, x5))))))))), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.55 = { by lemma 21 }
% 0.18/0.55 ifeq2(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(product(x3, x5), product(x2, product(x3, x5))))))), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.55 = { by lemma 16 }
% 0.18/0.55 ifeq2(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(x3, x5))))), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.55 = { by lemma 18 }
% 0.18/0.55 ifeq2(product(product(x3, x5), product(product(x2, x4), product(x3, x5))), product(x3, x5), ifeq2(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), d(product(x2, x4), product(x3, x5)), true), true)
% 0.18/0.55 = { by axiom 9 (sos03) }
% 0.18/0.55 true
% 0.18/0.55 % SZS output end Proof
% 0.18/0.55
% 0.18/0.55 RESULT: Theorem (the conjecture is true).
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