%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : ALG178+1 : TPTP v9.3.1. Released v2.7.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/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 09:07:07 AM UTC 2026
% Result : Theorem 0.20s 0.26s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : ALG178+1 : TPTP v9.3.1. Released v2.7.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.06/0.19 % Computer : n005.cluster.edu
% 0.06/0.19 % Model : x86_64 x86_64
% 0.06/0.19 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.19 % Memory : 8046.5625MB
% 0.06/0.19 % OS : Linux 6.8.0-71-generic
% 0.06/0.19 % CPULimit : 300
% 0.06/0.19 % WCLimit : 300
% 0.06/0.19 % DateTime : Mon Sep 28 19:38:31 UTC 2026
% 0.06/0.19 % CPUTime :
% 0.06/0.19 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.20/0.26 Command-line arguments: --flatten --complete-subsets
% 0.20/0.26
% 0.20/0.26 % SZS status Theorem
% 0.20/0.26
% 0.20/0.27 % SZS output start Proof
% 0.20/0.27 Axiom 1 (ax4_1): sorti2(v) = true.
% 0.20/0.27 Axiom 2 (ax4): sorti2(u) = true.
% 0.20/0.27 Axiom 3 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.20/0.27 Axiom 4 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 0.20/0.27 Axiom 5 (co1_3): ifeq(sorti2(X), true, sorti1(j(X)), true) = true.
% 0.20/0.27 Axiom 6 (co1_5): ifeq2(sorti2(X), true, h(j(X)), X) = X.
% 0.20/0.27 Axiom 7 (ax2): ifeq(sorti2(X), true, ifeq(sorti2(Y), true, sorti2(op2(Y, X)), true), true) = true.
% 0.20/0.27 Axiom 8 (ax3): ifeq2(sorti1(X), true, ifeq2(sorti1(Y), true, op1(Y, op1(Y, X)), X), X) = X.
% 0.20/0.27 Axiom 9 (co1_4): ifeq2(sorti2(X), true, ifeq2(sorti2(Y), true, op1(j(Y), j(X)), j(op2(Y, X))), j(op2(Y, X))) = j(op2(Y, X)).
% 0.20/0.27 Axiom 10 (co1_1): ifeq2(sorti1(X), true, ifeq2(sorti1(Y), true, op2(h(Y), h(X)), h(op1(Y, X))), h(op1(Y, X))) = h(op1(Y, X)).
% 0.20/0.27
% 0.20/0.27 Lemma 11: sorti2(u) = sorti2(v).
% 0.20/0.27 Proof:
% 0.20/0.27 sorti2(u)
% 0.20/0.27 = { by axiom 2 (ax4) }
% 0.20/0.27 true
% 0.20/0.27 = { by axiom 1 (ax4_1) R->L }
% 0.20/0.27 sorti2(v)
% 0.20/0.27
% 0.20/0.27 Lemma 12: ifeq(sorti2(X), sorti2(u), sorti1(j(X)), sorti2(u)) = sorti2(u).
% 0.20/0.27 Proof:
% 0.20/0.27 ifeq(sorti2(X), sorti2(u), sorti1(j(X)), sorti2(u))
% 0.20/0.27 = { by lemma 11 }
% 0.20/0.27 ifeq(sorti2(X), sorti2(v), sorti1(j(X)), sorti2(u))
% 0.20/0.27 = { by lemma 11 }
% 0.20/0.27 ifeq(sorti2(X), sorti2(v), sorti1(j(X)), sorti2(v))
% 0.20/0.27 = { by axiom 1 (ax4_1) }
% 0.20/0.27 ifeq(sorti2(X), true, sorti1(j(X)), sorti2(v))
% 0.20/0.27 = { by axiom 1 (ax4_1) }
% 0.20/0.27 ifeq(sorti2(X), true, sorti1(j(X)), true)
% 0.20/0.27 = { by axiom 5 (co1_3) }
% 0.20/0.27 true
% 0.20/0.27 = { by axiom 1 (ax4_1) R->L }
% 0.20/0.27 sorti2(v)
% 0.20/0.27 = { by lemma 11 R->L }
% 0.20/0.27 sorti2(u)
% 0.20/0.27
% 0.20/0.27 Lemma 13: sorti1(j(u)) = sorti2(u).
% 0.20/0.27 Proof:
% 0.20/0.27 sorti1(j(u))
% 0.20/0.28 = { by axiom 3 (ifeq_axiom) R->L }
% 0.20/0.28 ifeq(sorti2(u), sorti2(u), sorti1(j(u)), sorti2(u))
% 0.20/0.28 = { by lemma 12 }
% 0.20/0.28 sorti2(u)
% 0.20/0.28
% 0.20/0.28 Lemma 14: sorti2(op2(u, v)) = sorti2(u).
% 0.20/0.28 Proof:
% 0.20/0.28 sorti2(op2(u, v))
% 0.20/0.28 = { by axiom 3 (ifeq_axiom) R->L }
% 0.20/0.28 ifeq(sorti2(u), sorti2(u), sorti2(op2(u, v)), sorti2(u))
% 0.20/0.28 = { by axiom 3 (ifeq_axiom) R->L }
% 0.20/0.28 ifeq(sorti2(u), sorti2(u), ifeq(sorti2(u), sorti2(u), sorti2(op2(u, v)), sorti2(u)), sorti2(u))
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq(sorti2(v), sorti2(u), ifeq(sorti2(u), sorti2(u), sorti2(op2(u, v)), sorti2(u)), sorti2(u))
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq(sorti2(v), sorti2(v), ifeq(sorti2(u), sorti2(u), sorti2(op2(u, v)), sorti2(u)), sorti2(u))
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq(sorti2(v), sorti2(v), ifeq(sorti2(u), sorti2(v), sorti2(op2(u, v)), sorti2(u)), sorti2(u))
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq(sorti2(v), sorti2(v), ifeq(sorti2(u), sorti2(v), sorti2(op2(u, v)), sorti2(v)), sorti2(u))
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq(sorti2(v), sorti2(v), ifeq(sorti2(u), sorti2(v), sorti2(op2(u, v)), sorti2(v)), sorti2(v))
% 0.20/0.28 = { by axiom 1 (ax4_1) }
% 0.20/0.28 ifeq(sorti2(v), true, ifeq(sorti2(u), sorti2(v), sorti2(op2(u, v)), sorti2(v)), sorti2(v))
% 0.20/0.28 = { by axiom 1 (ax4_1) }
% 0.20/0.28 ifeq(sorti2(v), true, ifeq(sorti2(u), true, sorti2(op2(u, v)), sorti2(v)), sorti2(v))
% 0.20/0.28 = { by axiom 1 (ax4_1) }
% 0.20/0.28 ifeq(sorti2(v), true, ifeq(sorti2(u), true, sorti2(op2(u, v)), true), sorti2(v))
% 0.20/0.28 = { by axiom 1 (ax4_1) }
% 0.20/0.28 ifeq(sorti2(v), true, ifeq(sorti2(u), true, sorti2(op2(u, v)), true), true)
% 0.20/0.28 = { by axiom 7 (ax2) }
% 0.20/0.28 true
% 0.20/0.28 = { by axiom 1 (ax4_1) R->L }
% 0.20/0.28 sorti2(v)
% 0.20/0.28 = { by lemma 11 R->L }
% 0.20/0.28 sorti2(u)
% 0.20/0.28
% 0.20/0.28 Lemma 15: op1(j(u), j(op2(u, v))) = j(v).
% 0.20/0.28 Proof:
% 0.20/0.28 op1(j(u), j(op2(u, v)))
% 0.20/0.28 = { by axiom 4 (ifeq_axiom) R->L }
% 0.20/0.28 ifeq2(sorti2(u), sorti2(u), op1(j(u), j(op2(u, v))), j(v))
% 0.20/0.28 = { by lemma 13 R->L }
% 0.20/0.28 ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), j(op2(u, v))), j(v))
% 0.20/0.28 = { by axiom 4 (ifeq_axiom) R->L }
% 0.20/0.28 ifeq2(sorti2(u), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), j(op2(u, v))), j(v)), j(v))
% 0.20/0.28 = { by lemma 12 R->L }
% 0.20/0.28 ifeq2(ifeq(sorti2(v), sorti2(u), sorti1(j(v)), sorti2(u)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), j(op2(u, v))), j(v)), j(v))
% 0.20/0.28 = { by lemma 11 R->L }
% 0.20/0.28 ifeq2(ifeq(sorti2(u), sorti2(u), sorti1(j(v)), sorti2(u)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), j(op2(u, v))), j(v)), j(v))
% 0.20/0.28 = { by axiom 3 (ifeq_axiom) }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), j(op2(u, v))), j(v)), j(v))
% 0.20/0.28 = { by axiom 9 (co1_4) R->L }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), ifeq2(sorti2(v), true, ifeq2(sorti2(u), true, op1(j(u), j(v)), j(op2(u, v))), j(op2(u, v)))), j(v)), j(v))
% 0.20/0.28 = { by axiom 1 (ax4_1) R->L }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), ifeq2(sorti2(v), true, ifeq2(sorti2(u), sorti2(v), op1(j(u), j(v)), j(op2(u, v))), j(op2(u, v)))), j(v)), j(v))
% 0.20/0.28 = { by axiom 1 (ax4_1) R->L }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), ifeq2(sorti2(v), sorti2(v), ifeq2(sorti2(u), sorti2(v), op1(j(u), j(v)), j(op2(u, v))), j(op2(u, v)))), j(v)), j(v))
% 0.20/0.28 = { by lemma 11 R->L }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), ifeq2(sorti2(v), sorti2(v), ifeq2(sorti2(u), sorti2(u), op1(j(u), j(v)), j(op2(u, v))), j(op2(u, v)))), j(v)), j(v))
% 0.20/0.28 = { by lemma 11 R->L }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), ifeq2(sorti2(v), sorti2(u), ifeq2(sorti2(u), sorti2(u), op1(j(u), j(v)), j(op2(u, v))), j(op2(u, v)))), j(v)), j(v))
% 0.20/0.28 = { by lemma 11 R->L }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), ifeq2(sorti2(u), sorti2(u), ifeq2(sorti2(u), sorti2(u), op1(j(u), j(v)), j(op2(u, v))), j(op2(u, v)))), j(v)), j(v))
% 0.20/0.28 = { by axiom 4 (ifeq_axiom) }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), ifeq2(sorti2(u), sorti2(u), op1(j(u), j(v)), j(op2(u, v)))), j(v)), j(v))
% 0.20/0.28 = { by axiom 4 (ifeq_axiom) }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), op1(j(u), j(v))), j(v)), j(v))
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(v), ifeq2(sorti1(j(u)), sorti2(u), op1(j(u), op1(j(u), j(v))), j(v)), j(v))
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq2(sorti1(j(v)), sorti2(v), ifeq2(sorti1(j(u)), sorti2(v), op1(j(u), op1(j(u), j(v))), j(v)), j(v))
% 0.20/0.28 = { by axiom 1 (ax4_1) }
% 0.20/0.28 ifeq2(sorti1(j(v)), true, ifeq2(sorti1(j(u)), sorti2(v), op1(j(u), op1(j(u), j(v))), j(v)), j(v))
% 0.20/0.28 = { by axiom 1 (ax4_1) }
% 0.20/0.28 ifeq2(sorti1(j(v)), true, ifeq2(sorti1(j(u)), true, op1(j(u), op1(j(u), j(v))), j(v)), j(v))
% 0.20/0.28 = { by axiom 8 (ax3) }
% 0.20/0.28 j(v)
% 0.20/0.28
% 0.20/0.28 Lemma 16: ifeq2(sorti2(X), sorti2(u), h(j(X)), X) = X.
% 0.20/0.28 Proof:
% 0.20/0.28 ifeq2(sorti2(X), sorti2(u), h(j(X)), X)
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq2(sorti2(X), sorti2(v), h(j(X)), X)
% 0.20/0.28 = { by axiom 1 (ax4_1) }
% 0.20/0.28 ifeq2(sorti2(X), true, h(j(X)), X)
% 0.20/0.28 = { by axiom 6 (co1_5) }
% 0.20/0.28 X
% 0.20/0.28
% 0.20/0.28 Goal 1 (ax4_2): op2(u, op2(u, v)) = v.
% 0.20/0.28 Proof:
% 0.20/0.28 op2(u, op2(u, v))
% 0.20/0.28 = { by lemma 16 R->L }
% 0.20/0.28 op2(u, ifeq2(sorti2(op2(u, v)), sorti2(u), h(j(op2(u, v))), op2(u, v)))
% 0.20/0.28 = { by lemma 14 }
% 0.20/0.28 op2(u, ifeq2(sorti2(u), sorti2(u), h(j(op2(u, v))), op2(u, v)))
% 0.20/0.28 = { by axiom 4 (ifeq_axiom) }
% 0.20/0.28 op2(u, h(j(op2(u, v))))
% 0.20/0.28 = { by lemma 16 R->L }
% 0.20/0.28 op2(ifeq2(sorti2(u), sorti2(u), h(j(u)), u), h(j(op2(u, v))))
% 0.20/0.28 = { by axiom 4 (ifeq_axiom) }
% 0.20/0.28 op2(h(j(u)), h(j(op2(u, v))))
% 0.20/0.28 = { by axiom 4 (ifeq_axiom) R->L }
% 0.20/0.28 ifeq2(sorti2(u), sorti2(u), op2(h(j(u)), h(j(op2(u, v)))), h(j(v)))
% 0.20/0.28 = { by lemma 13 R->L }
% 0.20/0.28 ifeq2(sorti1(j(u)), sorti2(u), op2(h(j(u)), h(j(op2(u, v)))), h(j(v)))
% 0.20/0.28 = { by axiom 4 (ifeq_axiom) R->L }
% 0.20/0.28 ifeq2(sorti2(u), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op2(h(j(u)), h(j(op2(u, v)))), h(j(v))), h(op1(j(u), j(op2(u, v)))))
% 0.20/0.28 = { by lemma 12 R->L }
% 0.20/0.28 ifeq2(ifeq(sorti2(op2(u, v)), sorti2(u), sorti1(j(op2(u, v))), sorti2(u)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op2(h(j(u)), h(j(op2(u, v)))), h(j(v))), h(op1(j(u), j(op2(u, v)))))
% 0.20/0.28 = { by lemma 14 }
% 0.20/0.28 ifeq2(ifeq(sorti2(u), sorti2(u), sorti1(j(op2(u, v))), sorti2(u)), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op2(h(j(u)), h(j(op2(u, v)))), h(j(v))), h(op1(j(u), j(op2(u, v)))))
% 0.20/0.28 = { by axiom 3 (ifeq_axiom) }
% 0.20/0.28 ifeq2(sorti1(j(op2(u, v))), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op2(h(j(u)), h(j(op2(u, v)))), h(j(v))), h(op1(j(u), j(op2(u, v)))))
% 0.20/0.28 = { by lemma 15 R->L }
% 0.20/0.28 ifeq2(sorti1(j(op2(u, v))), sorti2(u), ifeq2(sorti1(j(u)), sorti2(u), op2(h(j(u)), h(j(op2(u, v)))), h(op1(j(u), j(op2(u, v))))), h(op1(j(u), j(op2(u, v)))))
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq2(sorti1(j(op2(u, v))), sorti2(v), ifeq2(sorti1(j(u)), sorti2(u), op2(h(j(u)), h(j(op2(u, v)))), h(op1(j(u), j(op2(u, v))))), h(op1(j(u), j(op2(u, v)))))
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq2(sorti1(j(op2(u, v))), sorti2(v), ifeq2(sorti1(j(u)), sorti2(v), op2(h(j(u)), h(j(op2(u, v)))), h(op1(j(u), j(op2(u, v))))), h(op1(j(u), j(op2(u, v)))))
% 0.20/0.28 = { by axiom 1 (ax4_1) }
% 0.20/0.28 ifeq2(sorti1(j(op2(u, v))), true, ifeq2(sorti1(j(u)), sorti2(v), op2(h(j(u)), h(j(op2(u, v)))), h(op1(j(u), j(op2(u, v))))), h(op1(j(u), j(op2(u, v)))))
% 0.20/0.28 = { by axiom 1 (ax4_1) }
% 0.20/0.28 ifeq2(sorti1(j(op2(u, v))), true, ifeq2(sorti1(j(u)), true, op2(h(j(u)), h(j(op2(u, v)))), h(op1(j(u), j(op2(u, v))))), h(op1(j(u), j(op2(u, v)))))
% 0.20/0.28 = { by axiom 10 (co1_1) }
% 0.20/0.28 h(op1(j(u), j(op2(u, v))))
% 0.20/0.28 = { by lemma 15 }
% 0.20/0.28 h(j(v))
% 0.20/0.28 = { by axiom 4 (ifeq_axiom) R->L }
% 0.20/0.28 ifeq2(sorti2(u), sorti2(u), h(j(v)), v)
% 0.20/0.28 = { by lemma 11 }
% 0.20/0.28 ifeq2(sorti2(v), sorti2(u), h(j(v)), v)
% 0.20/0.28 = { by lemma 16 }
% 0.20/0.28 v
% 0.20/0.28 % SZS output end Proof
% 0.20/0.28
% 0.20/0.28 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------