%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : ALG210+2 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n004.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
% 0.00/0.02 % Problem : ALG210+2 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.07/0.18 % Computer : n004.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 19:43:08 UTC 2026
% 0.07/0.18 % CPUTime :
% 0.07/0.18 Running run_twee /export/starexec/sandbox/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
% 0.21/0.32 % SZS output start Proof
% 0.21/0.32 Axiom 1 (conjecture_1_1): element(b) = true.
% 0.21/0.32 Axiom 2 (conjecture_1_2): element(a) = true.
% 0.21/0.32 Axiom 3 (conjecture_1): c = times(a, b).
% 0.21/0.32 Axiom 4 (axiom_1): times(times(X, Y), Z) = times(Y, times(Z, X)).
% 0.21/0.32 Axiom 5 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.21/0.32 Axiom 6 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 0.21/0.32 Axiom 7 (axiom_2_2): ifeq(element(X), true, times(X, X), c2(X)) = c2(X).
% 0.21/0.32 Axiom 8 (axiom_2_1): ifeq(element(X), true, times(X, c2(X)), X) = X.
% 0.21/0.32 Axiom 9 (axiom_2): ifeq2(times(X, X), Y, ifeq2(times(X, Y), X, element(X), true), true) = true.
% 0.21/0.32
% 0.21/0.32 Lemma 10: times(b, b) = c2(b).
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, b)
% 0.21/0.32 = { by axiom 5 (ifeq_axiom) R->L }
% 0.21/0.32 ifeq(true, true, times(b, b), c2(b))
% 0.21/0.32 = { by axiom 1 (conjecture_1_1) R->L }
% 0.21/0.32 ifeq(element(b), true, times(b, b), c2(b))
% 0.21/0.32 = { by axiom 7 (axiom_2_2) }
% 0.21/0.32 c2(b)
% 0.21/0.32
% 0.21/0.32 Lemma 11: times(b, c2(b)) = b.
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, c2(b))
% 0.21/0.32 = { by axiom 5 (ifeq_axiom) R->L }
% 0.21/0.32 ifeq(true, true, times(b, c2(b)), b)
% 0.21/0.32 = { by axiom 1 (conjecture_1_1) R->L }
% 0.21/0.32 ifeq(element(b), true, times(b, c2(b)), b)
% 0.21/0.32 = { by axiom 8 (axiom_2_1) }
% 0.21/0.32 b
% 0.21/0.32
% 0.21/0.32 Lemma 12: times(c2(b), c) = times(b, a).
% 0.21/0.32 Proof:
% 0.21/0.32 times(c2(b), c)
% 0.21/0.32 = { by axiom 3 (conjecture_1) }
% 0.21/0.32 times(c2(b), times(a, b))
% 0.21/0.32 = { by axiom 4 (axiom_1) R->L }
% 0.21/0.32 times(times(b, c2(b)), a)
% 0.21/0.32 = { by lemma 11 }
% 0.21/0.32 times(b, a)
% 0.21/0.32
% 0.21/0.32 Lemma 13: times(b, times(X, a)) = times(c, X).
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, times(X, a))
% 0.21/0.32 = { by axiom 4 (axiom_1) R->L }
% 0.21/0.32 times(times(a, b), X)
% 0.21/0.32 = { by axiom 3 (conjecture_1) R->L }
% 0.21/0.32 times(c, X)
% 0.21/0.32
% 0.21/0.32 Lemma 14: times(X, times(Y, times(Z, W))) = times(Z, times(W, times(X, Y))).
% 0.21/0.32 Proof:
% 0.21/0.32 times(X, times(Y, times(Z, W)))
% 0.21/0.32 = { by axiom 4 (axiom_1) R->L }
% 0.21/0.32 times(times(times(Z, W), X), Y)
% 0.21/0.32 = { by axiom 4 (axiom_1) }
% 0.21/0.32 times(times(W, times(X, Z)), Y)
% 0.21/0.32 = { by axiom 4 (axiom_1) }
% 0.21/0.32 times(times(X, Z), times(Y, W))
% 0.21/0.32 = { by axiom 4 (axiom_1) }
% 0.21/0.32 times(Z, times(times(Y, W), X))
% 0.21/0.32 = { by axiom 4 (axiom_1) }
% 0.21/0.32 times(Z, times(W, times(X, Y)))
% 0.21/0.32
% 0.21/0.32 Lemma 15: times(a, times(b, times(X, Y))) = times(X, times(Y, c)).
% 0.21/0.32 Proof:
% 0.21/0.32 times(a, times(b, times(X, Y)))
% 0.21/0.32 = { by lemma 14 R->L }
% 0.21/0.32 times(X, times(Y, times(a, b)))
% 0.21/0.32 = { by axiom 3 (conjecture_1) R->L }
% 0.21/0.32 times(X, times(Y, c))
% 0.21/0.32
% 0.21/0.32 Lemma 16: times(a, times(c, X)) = times(X, times(a, c)).
% 0.21/0.32 Proof:
% 0.21/0.32 times(a, times(c, X))
% 0.21/0.32 = { by lemma 13 R->L }
% 0.21/0.32 times(a, times(b, times(X, a)))
% 0.21/0.32 = { by lemma 15 }
% 0.21/0.32 times(X, times(a, c))
% 0.21/0.32
% 0.21/0.32 Lemma 17: times(b, times(X, c)) = times(c, times(b, X)).
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, times(X, c))
% 0.21/0.32 = { by axiom 3 (conjecture_1) }
% 0.21/0.32 times(b, times(X, times(a, b)))
% 0.21/0.32 = { by axiom 4 (axiom_1) R->L }
% 0.21/0.32 times(b, times(times(b, X), a))
% 0.21/0.32 = { by lemma 13 }
% 0.21/0.32 times(c, times(b, X))
% 0.21/0.32
% 0.21/0.32 Lemma 18: times(b, times(X, b)) = times(c2(b), X).
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, times(X, b))
% 0.21/0.32 = { by axiom 4 (axiom_1) R->L }
% 0.21/0.32 times(times(b, b), X)
% 0.21/0.32 = { by lemma 10 }
% 0.21/0.32 times(c2(b), X)
% 0.21/0.32
% 0.21/0.32 Lemma 19: times(b, times(b, c)) = c.
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, times(b, c))
% 0.21/0.32 = { by lemma 15 R->L }
% 0.21/0.32 times(a, times(b, times(b, b)))
% 0.21/0.32 = { by lemma 10 }
% 0.21/0.32 times(a, times(b, c2(b)))
% 0.21/0.32 = { by lemma 11 }
% 0.21/0.32 times(a, b)
% 0.21/0.32 = { by axiom 3 (conjecture_1) R->L }
% 0.21/0.32 c
% 0.21/0.32
% 0.21/0.32 Lemma 20: times(X, times(a, times(Y, b))) = times(Y, times(c, X)).
% 0.21/0.32 Proof:
% 0.21/0.32 times(X, times(a, times(Y, b)))
% 0.21/0.32 = { by lemma 14 R->L }
% 0.21/0.32 times(Y, times(b, times(X, a)))
% 0.21/0.32 = { by lemma 13 }
% 0.21/0.32 times(Y, times(c, X))
% 0.21/0.32
% 0.21/0.32 Lemma 21: times(b, times(b, times(X, Y))) = times(X, times(Y, c2(b))).
% 0.21/0.32 Proof:
% 0.21/0.32 times(b, times(b, times(X, Y)))
% 0.21/0.32 = { by lemma 14 R->L }
% 0.21/0.32 times(X, times(Y, times(b, b)))
% 0.21/0.32 = { by lemma 10 }
% 0.21/0.32 times(X, times(Y, c2(b)))
% 0.21/0.32
% 0.21/0.32 Goal 1 (conjecture_1_3): element(c) = true.
% 0.21/0.32 Proof:
% 0.21/0.32 element(c)
% 0.21/0.32 = { by axiom 6 (ifeq_axiom) R->L }
% 0.21/0.32 ifeq2(c, c, element(c), true)
% 0.21/0.32 = { by axiom 6 (ifeq_axiom) R->L }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(c, c, element(c), true), true)
% 0.21/0.32 = { by axiom 3 (conjecture_1) }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(a, b), c, element(c), true), true)
% 0.21/0.32 = { by axiom 8 (axiom_2_1) R->L }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(ifeq(element(a), true, times(a, c2(a)), a), b), c, element(c), true), true)
% 0.21/0.32 = { by axiom 2 (conjecture_1_2) }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(ifeq(true, true, times(a, c2(a)), a), b), c, element(c), true), true)
% 0.21/0.32 = { by axiom 5 (ifeq_axiom) }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(times(a, c2(a)), b), c, element(c), true), true)
% 0.21/0.32 = { by axiom 4 (axiom_1) }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), times(b, a)), c, element(c), true), true)
% 0.21/0.32 = { by lemma 12 R->L }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), times(c2(b), c)), c, element(c), true), true)
% 0.21/0.32 = { by lemma 18 R->L }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), times(b, times(c, b))), c, element(c), true), true)
% 0.21/0.32 = { by lemma 11 R->L }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), times(b, times(c, times(b, c2(b))))), c, element(c), true), true)
% 0.21/0.32 = { by lemma 10 R->L }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), times(b, times(c, times(b, times(b, b))))), c, element(c), true), true)
% 0.21/0.32 = { by lemma 18 }
% 0.21/0.32 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), times(b, times(c, times(c2(b), b)))), c, element(c), true), true)
% 0.21/0.32 = { by axiom 4 (axiom_1) R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), times(b, times(times(b, c), c2(b)))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 21 R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), times(b, times(b, times(b, times(b, c))))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 19 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), times(b, times(b, c))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 19 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c2(a), c), c, element(c), true), true)
% 0.21/0.33 = { by axiom 7 (axiom_2_2) R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(ifeq(element(a), true, times(a, a), c2(a)), c), c, element(c), true), true)
% 0.21/0.33 = { by axiom 2 (conjecture_1_2) }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(ifeq(true, true, times(a, a), c2(a)), c), c, element(c), true), true)
% 0.21/0.33 = { by axiom 5 (ifeq_axiom) }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(times(a, a), c), c, element(c), true), true)
% 0.21/0.33 = { by axiom 4 (axiom_1) }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(a, times(c, a)), c, element(c), true), true)
% 0.21/0.33 = { by lemma 13 R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(a, times(b, times(a, a))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 15 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(a, times(a, c)), c, element(c), true), true)
% 0.21/0.33 = { by lemma 19 R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(a, times(a, times(b, times(b, c)))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 17 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(a, times(a, times(c, times(b, b)))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 10 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(a, times(a, times(c, c2(b)))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 21 R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(a, times(b, times(b, times(a, c)))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 16 R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(a, times(b, times(a, times(c, b)))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 20 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(a, times(c, times(c, b))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 16 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(times(c, b), times(a, c)), c, element(c), true), true)
% 0.21/0.33 = { by axiom 4 (axiom_1) }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(b, times(times(a, c), c)), c, element(c), true), true)
% 0.21/0.33 = { by lemma 17 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c, times(b, times(a, c))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 16 R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c, times(a, times(c, b))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 13 R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c, times(a, times(b, times(b, a)))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 12 R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c, times(a, times(b, times(c2(b), c)))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 15 R->L }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c, times(a, times(a, times(b, times(b, c2(b)))))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 11 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c, times(a, times(a, times(b, b)))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 20 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c, times(b, times(c, a))), c, element(c), true), true)
% 0.21/0.33 = { by lemma 13 }
% 0.21/0.33 ifeq2(times(c, c), times(c, c), ifeq2(times(c, times(c, c)), c, element(c), true), true)
% 0.21/0.33 = { by axiom 9 (axiom_2) }
% 0.21/0.33 true
% 0.21/0.33 % SZS output end Proof
% 0.21/0.33
% 0.21/0.33 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------