%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : ALG210+1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n005.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 09:07:09 AM UTC 2026
% Result : Theorem 0.21s 0.29s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : ALG210+1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.07/0.18 % Computer : n005.cluster.edu
% 0.07/0.18 % Model : x86_64 x86_64
% 0.07/0.18 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.18 % Memory : 8046.5625MB
% 0.07/0.18 % OS : Linux 6.8.0-71-generic
% 0.07/0.18 % CPULimit : 300
% 0.07/0.18 % WCLimit : 300
% 0.07/0.18 % DateTime : Mon Sep 28 19:42:01 UTC 2026
% 0.07/0.19 % CPUTime :
% 0.07/0.19 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.21/0.29 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-every 2
% 0.21/0.29
% 0.21/0.29 % SZS status Theorem
% 0.21/0.29
% 0.21/0.31 % SZS output start Proof
% 0.21/0.31 Axiom 1 (conjecture_1): element(b) = true.
% 0.21/0.31 Axiom 2 (conjecture_1_1): element(a) = true.
% 0.21/0.31 Axiom 3 (axiom_1): times(times(X, Y), Z) = times(Y, times(Z, X)).
% 0.21/0.31 Axiom 4 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.21/0.31 Axiom 5 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 0.21/0.31 Axiom 6 (axiom_2_2): ifeq(element(X), true, times(X, X), c(X)) = c(X).
% 0.21/0.31 Axiom 7 (axiom_2_1): ifeq(element(X), true, times(X, c(X)), X) = X.
% 0.21/0.31 Axiom 8 (axiom_2): ifeq2(times(X, X), Y, ifeq2(times(X, Y), X, element(X), true), true) = true.
% 0.21/0.31
% 0.21/0.31 Lemma 9: times(b, c(b)) = b.
% 0.21/0.31 Proof:
% 0.21/0.31 times(b, c(b))
% 0.21/0.31 = { by axiom 4 (ifeq_axiom) R->L }
% 0.21/0.31 ifeq(true, true, times(b, c(b)), b)
% 0.21/0.31 = { by axiom 1 (conjecture_1) R->L }
% 0.21/0.31 ifeq(element(b), true, times(b, c(b)), b)
% 0.21/0.31 = { by axiom 7 (axiom_2_1) }
% 0.21/0.31 b
% 0.21/0.31
% 0.21/0.31 Lemma 10: times(c(b), times(X, b)) = times(b, X).
% 0.21/0.31 Proof:
% 0.21/0.31 times(c(b), times(X, b))
% 0.21/0.31 = { by axiom 3 (axiom_1) R->L }
% 0.21/0.31 times(times(b, c(b)), X)
% 0.21/0.31 = { by lemma 9 }
% 0.21/0.31 times(b, X)
% 0.21/0.31
% 0.21/0.31 Lemma 11: times(b, b) = c(b).
% 0.21/0.31 Proof:
% 0.21/0.31 times(b, b)
% 0.21/0.31 = { by axiom 4 (ifeq_axiom) R->L }
% 0.21/0.31 ifeq(true, true, times(b, b), c(b))
% 0.21/0.31 = { by axiom 1 (conjecture_1) R->L }
% 0.21/0.31 ifeq(element(b), true, times(b, b), c(b))
% 0.21/0.31 = { by axiom 6 (axiom_2_2) }
% 0.21/0.31 c(b)
% 0.21/0.31
% 0.21/0.31 Lemma 12: times(b, times(X, b)) = times(c(b), X).
% 0.21/0.31 Proof:
% 0.21/0.31 times(b, times(X, b))
% 0.21/0.31 = { by axiom 3 (axiom_1) R->L }
% 0.21/0.31 times(times(b, b), X)
% 0.21/0.31 = { by lemma 11 }
% 0.21/0.31 times(c(b), X)
% 0.21/0.31
% 0.21/0.31 Lemma 13: times(X, times(Y, times(Z, W))) = times(Z, times(W, times(X, Y))).
% 0.21/0.31 Proof:
% 0.21/0.31 times(X, times(Y, times(Z, W)))
% 0.21/0.31 = { by axiom 3 (axiom_1) R->L }
% 0.21/0.32 times(times(times(Z, W), X), Y)
% 0.21/0.32 = { by axiom 3 (axiom_1) }
% 0.21/0.32 times(times(W, times(X, Z)), Y)
% 0.21/0.32 = { by axiom 3 (axiom_1) }
% 0.21/0.32 times(times(X, Z), times(Y, W))
% 0.21/0.32 = { by axiom 3 (axiom_1) }
% 0.21/0.32 times(Z, times(times(Y, W), X))
% 0.21/0.32 = { by axiom 3 (axiom_1) }
% 0.21/0.32 times(Z, times(W, times(X, Y)))
% 0.21/0.32
% 0.21/0.32 Lemma 14: times(b, times(b, times(X, Y))) = times(X, times(Y, c(b))).
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, times(b, times(X, Y)))
% 0.21/0.32 = { by lemma 13 R->L }
% 0.21/0.32 times(X, times(Y, times(b, b)))
% 0.21/0.32 = { by lemma 11 }
% 0.21/0.32 times(X, times(Y, c(b)))
% 0.21/0.32
% 0.21/0.32 Lemma 15: times(b, times(b, times(X, b))) = times(X, b).
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, times(b, times(X, b)))
% 0.21/0.32 = { by lemma 13 }
% 0.21/0.32 times(X, times(b, times(b, b)))
% 0.21/0.32 = { by lemma 11 }
% 0.21/0.32 times(X, times(b, c(b)))
% 0.21/0.32 = { by lemma 9 }
% 0.21/0.32 times(X, b)
% 0.21/0.32
% 0.21/0.32 Lemma 16: times(b, X) = times(X, b).
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, X)
% 0.21/0.32 = { by lemma 10 R->L }
% 0.21/0.32 times(c(b), times(X, b))
% 0.21/0.32 = { by lemma 12 R->L }
% 0.21/0.32 times(b, times(times(X, b), b))
% 0.21/0.32 = { by lemma 9 R->L }
% 0.21/0.32 times(b, times(times(X, b), times(b, c(b))))
% 0.21/0.32 = { by lemma 11 R->L }
% 0.21/0.32 times(b, times(times(X, b), times(b, times(b, b))))
% 0.21/0.32 = { by lemma 12 }
% 0.21/0.32 times(b, times(times(X, b), times(c(b), b)))
% 0.21/0.32 = { by axiom 3 (axiom_1) R->L }
% 0.21/0.32 times(b, times(times(b, times(X, b)), c(b)))
% 0.21/0.32 = { by lemma 14 R->L }
% 0.21/0.32 times(b, times(b, times(b, times(b, times(X, b)))))
% 0.21/0.32 = { by lemma 15 }
% 0.21/0.32 times(b, times(b, times(X, b)))
% 0.21/0.32 = { by lemma 15 }
% 0.21/0.32 times(X, b)
% 0.21/0.32
% 0.21/0.32 Lemma 17: times(X, times(Y, times(Z, W))) = times(Z, times(times(Y, W), X)).
% 0.21/0.32 Proof:
% 0.21/0.32 times(X, times(Y, times(Z, W)))
% 0.21/0.32 = { by lemma 13 }
% 0.21/0.32 times(Z, times(W, times(X, Y)))
% 0.21/0.32 = { by axiom 3 (axiom_1) R->L }
% 0.21/0.32 times(Z, times(times(Y, W), X))
% 0.21/0.32
% 0.21/0.32 Goal 1 (conjecture_1_2): element(times(a, b)) = true.
% 0.21/0.32 Proof:
% 0.21/0.32 element(times(a, b))
% 0.21/0.32 = { by axiom 5 (ifeq_axiom) R->L }
% 0.21/0.32 ifeq2(times(a, b), times(a, b), element(times(a, b)), true)
% 0.21/0.32 = { by axiom 5 (ifeq_axiom) R->L }
% 0.21/0.32 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, b), times(a, b), element(times(a, b)), true), true)
% 0.21/0.32 = { by lemma 16 R->L }
% 0.21/0.32 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(b, a), times(a, b), element(times(a, b)), true), true)
% 0.21/0.32 = { by lemma 10 R->L }
% 0.21/0.32 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(c(b), times(a, b)), times(a, b), element(times(a, b)), true), true)
% 0.21/0.32 = { by axiom 7 (axiom_2_1) R->L }
% 0.21/0.32 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(c(b), times(ifeq(element(a), true, times(a, c(a)), a), b)), times(a, b), element(times(a, b)), true), true)
% 0.21/0.32 = { by axiom 2 (conjecture_1_1) }
% 0.21/0.32 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(c(b), times(ifeq(true, true, times(a, c(a)), a), b)), times(a, b), element(times(a, b)), true), true)
% 0.21/0.32 = { by axiom 4 (ifeq_axiom) }
% 0.21/0.32 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(c(b), times(times(a, c(a)), b)), times(a, b), element(times(a, b)), true), true)
% 0.21/0.32 = { by axiom 3 (axiom_1) }
% 0.21/0.32 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(c(b), times(c(a), times(b, a))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 16 }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(c(b), times(c(a), times(a, b))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 13 }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(c(b), c(a)))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 12 R->L }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(c(a), b)))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 9 R->L }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(c(a), times(b, c(b)))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 14 R->L }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(b, times(b, times(c(a), b)))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 12 }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(b, times(c(b), c(a)))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by axiom 6 (axiom_2_2) R->L }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(b, times(c(b), ifeq(element(a), true, times(a, a), c(a))))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by axiom 2 (conjecture_1_1) }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(b, times(c(b), ifeq(true, true, times(a, a), c(a))))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by axiom 4 (ifeq_axiom) }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(b, times(c(b), times(a, a)))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 13 }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(a, times(a, times(b, c(b))))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 9 }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(a, times(a, b))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 17 }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(a, times(times(a, b), b)))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by axiom 3 (axiom_1) }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(a, times(b, times(b, a))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 10 R->L }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(a, times(b, times(c(b), times(a, b)))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 13 }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(a, times(a, times(b, times(b, c(b))))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 9 }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(a, times(a, times(b, b))))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 17 }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(b, times(times(a, b), a)))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by axiom 3 (axiom_1) R->L }
% 0.21/0.33 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(a, times(b, times(times(a, b), times(a, b)))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.33 = { by lemma 13 R->L }
% 0.21/0.34 ifeq2(times(times(a, b), times(a, b)), times(times(a, b), times(a, b)), ifeq2(times(times(a, b), times(times(a, b), times(a, b))), times(a, b), element(times(a, b)), true), true)
% 0.21/0.34 = { by axiom 8 (axiom_2) }
% 0.21/0.34 true
% 0.21/0.34 % SZS output end Proof
% 0.21/0.34
% 0.21/0.34 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------