MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__U11(X1, X2, X3) -> U11(X1, X2, X3) , a__U11(tt(), M, N) -> a__U12(tt(), M, N) , a__U12(X1, X2, X3) -> U12(X1, X2, X3) , a__U12(tt(), M, N) -> s(a__plus(mark(N), mark(M))) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, s(M)) -> a__U11(tt(), M, N) , a__plus(N, 0()) -> mark(N) , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) , mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3) , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) , a__U21(X1, X2, X3) -> U21(X1, X2, X3) , a__U21(tt(), M, N) -> a__U22(tt(), M, N) , a__U22(X1, X2, X3) -> U22(X1, X2, X3) , a__U22(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) , a__x(X1, X2) -> x(X1, X2) , a__x(N, s(M)) -> a__U21(tt(), M, N) , a__x(N, 0()) -> 0() } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__U11^#(tt(), M, N) -> c_2(a__U12^#(tt(), M, N)) , a__U12^#(X1, X2, X3) -> c_3(X1, X2, X3) , a__U12^#(tt(), M, N) -> c_4(a__plus^#(mark(N), mark(M))) , a__plus^#(X1, X2) -> c_5(X1, X2) , a__plus^#(N, s(M)) -> c_6(a__U11^#(tt(), M, N)) , a__plus^#(N, 0()) -> c_7(mark^#(N)) , mark^#(tt()) -> c_8() , mark^#(s(X)) -> c_9(mark^#(X)) , mark^#(0()) -> c_10() , mark^#(U11(X1, X2, X3)) -> c_11(a__U11^#(mark(X1), X2, X3)) , mark^#(U12(X1, X2, X3)) -> c_12(a__U12^#(mark(X1), X2, X3)) , mark^#(plus(X1, X2)) -> c_13(a__plus^#(mark(X1), mark(X2))) , mark^#(U21(X1, X2, X3)) -> c_14(a__U21^#(mark(X1), X2, X3)) , mark^#(U22(X1, X2, X3)) -> c_15(a__U22^#(mark(X1), X2, X3)) , mark^#(x(X1, X2)) -> c_16(a__x^#(mark(X1), mark(X2))) , a__U21^#(X1, X2, X3) -> c_17(X1, X2, X3) , a__U21^#(tt(), M, N) -> c_18(a__U22^#(tt(), M, N)) , a__U22^#(X1, X2, X3) -> c_19(X1, X2, X3) , a__U22^#(tt(), M, N) -> c_20(a__plus^#(a__x(mark(N), mark(M)), mark(N))) , a__x^#(X1, X2) -> c_21(X1, X2) , a__x^#(N, s(M)) -> c_22(a__U21^#(tt(), M, N)) , a__x^#(N, 0()) -> c_23() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__U11^#(tt(), M, N) -> c_2(a__U12^#(tt(), M, N)) , a__U12^#(X1, X2, X3) -> c_3(X1, X2, X3) , a__U12^#(tt(), M, N) -> c_4(a__plus^#(mark(N), mark(M))) , a__plus^#(X1, X2) -> c_5(X1, X2) , a__plus^#(N, s(M)) -> c_6(a__U11^#(tt(), M, N)) , a__plus^#(N, 0()) -> c_7(mark^#(N)) , mark^#(tt()) -> c_8() , mark^#(s(X)) -> c_9(mark^#(X)) , mark^#(0()) -> c_10() , mark^#(U11(X1, X2, X3)) -> c_11(a__U11^#(mark(X1), X2, X3)) , mark^#(U12(X1, X2, X3)) -> c_12(a__U12^#(mark(X1), X2, X3)) , mark^#(plus(X1, X2)) -> c_13(a__plus^#(mark(X1), mark(X2))) , mark^#(U21(X1, X2, X3)) -> c_14(a__U21^#(mark(X1), X2, X3)) , mark^#(U22(X1, X2, X3)) -> c_15(a__U22^#(mark(X1), X2, X3)) , mark^#(x(X1, X2)) -> c_16(a__x^#(mark(X1), mark(X2))) , a__U21^#(X1, X2, X3) -> c_17(X1, X2, X3) , a__U21^#(tt(), M, N) -> c_18(a__U22^#(tt(), M, N)) , a__U22^#(X1, X2, X3) -> c_19(X1, X2, X3) , a__U22^#(tt(), M, N) -> c_20(a__plus^#(a__x(mark(N), mark(M)), mark(N))) , a__x^#(X1, X2) -> c_21(X1, X2) , a__x^#(N, s(M)) -> c_22(a__U21^#(tt(), M, N)) , a__x^#(N, 0()) -> c_23() } Strict Trs: { a__U11(X1, X2, X3) -> U11(X1, X2, X3) , a__U11(tt(), M, N) -> a__U12(tt(), M, N) , a__U12(X1, X2, X3) -> U12(X1, X2, X3) , a__U12(tt(), M, N) -> s(a__plus(mark(N), mark(M))) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, s(M)) -> a__U11(tt(), M, N) , a__plus(N, 0()) -> mark(N) , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) , mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3) , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) , a__U21(X1, X2, X3) -> U21(X1, X2, X3) , a__U21(tt(), M, N) -> a__U22(tt(), M, N) , a__U22(X1, X2, X3) -> U22(X1, X2, X3) , a__U22(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) , a__x(X1, X2) -> x(X1, X2) , a__x(N, s(M)) -> a__U21(tt(), M, N) , a__x(N, 0()) -> 0() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {8,10,23} by applications of Pre({8,10,23}) = {1,3,5,7,9,16,17,19,21}. Here rules are labeled as follows: DPs: { 1: a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) , 2: a__U11^#(tt(), M, N) -> c_2(a__U12^#(tt(), M, N)) , 3: a__U12^#(X1, X2, X3) -> c_3(X1, X2, X3) , 4: a__U12^#(tt(), M, N) -> c_4(a__plus^#(mark(N), mark(M))) , 5: a__plus^#(X1, X2) -> c_5(X1, X2) , 6: a__plus^#(N, s(M)) -> c_6(a__U11^#(tt(), M, N)) , 7: a__plus^#(N, 0()) -> c_7(mark^#(N)) , 8: mark^#(tt()) -> c_8() , 9: mark^#(s(X)) -> c_9(mark^#(X)) , 10: mark^#(0()) -> c_10() , 11: mark^#(U11(X1, X2, X3)) -> c_11(a__U11^#(mark(X1), X2, X3)) , 12: mark^#(U12(X1, X2, X3)) -> c_12(a__U12^#(mark(X1), X2, X3)) , 13: mark^#(plus(X1, X2)) -> c_13(a__plus^#(mark(X1), mark(X2))) , 14: mark^#(U21(X1, X2, X3)) -> c_14(a__U21^#(mark(X1), X2, X3)) , 15: mark^#(U22(X1, X2, X3)) -> c_15(a__U22^#(mark(X1), X2, X3)) , 16: mark^#(x(X1, X2)) -> c_16(a__x^#(mark(X1), mark(X2))) , 17: a__U21^#(X1, X2, X3) -> c_17(X1, X2, X3) , 18: a__U21^#(tt(), M, N) -> c_18(a__U22^#(tt(), M, N)) , 19: a__U22^#(X1, X2, X3) -> c_19(X1, X2, X3) , 20: a__U22^#(tt(), M, N) -> c_20(a__plus^#(a__x(mark(N), mark(M)), mark(N))) , 21: a__x^#(X1, X2) -> c_21(X1, X2) , 22: a__x^#(N, s(M)) -> c_22(a__U21^#(tt(), M, N)) , 23: a__x^#(N, 0()) -> c_23() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__U11^#(tt(), M, N) -> c_2(a__U12^#(tt(), M, N)) , a__U12^#(X1, X2, X3) -> c_3(X1, X2, X3) , a__U12^#(tt(), M, N) -> c_4(a__plus^#(mark(N), mark(M))) , a__plus^#(X1, X2) -> c_5(X1, X2) , a__plus^#(N, s(M)) -> c_6(a__U11^#(tt(), M, N)) , a__plus^#(N, 0()) -> c_7(mark^#(N)) , mark^#(s(X)) -> c_9(mark^#(X)) , mark^#(U11(X1, X2, X3)) -> c_11(a__U11^#(mark(X1), X2, X3)) , mark^#(U12(X1, X2, X3)) -> c_12(a__U12^#(mark(X1), X2, X3)) , mark^#(plus(X1, X2)) -> c_13(a__plus^#(mark(X1), mark(X2))) , mark^#(U21(X1, X2, X3)) -> c_14(a__U21^#(mark(X1), X2, X3)) , mark^#(U22(X1, X2, X3)) -> c_15(a__U22^#(mark(X1), X2, X3)) , mark^#(x(X1, X2)) -> c_16(a__x^#(mark(X1), mark(X2))) , a__U21^#(X1, X2, X3) -> c_17(X1, X2, X3) , a__U21^#(tt(), M, N) -> c_18(a__U22^#(tt(), M, N)) , a__U22^#(X1, X2, X3) -> c_19(X1, X2, X3) , a__U22^#(tt(), M, N) -> c_20(a__plus^#(a__x(mark(N), mark(M)), mark(N))) , a__x^#(X1, X2) -> c_21(X1, X2) , a__x^#(N, s(M)) -> c_22(a__U21^#(tt(), M, N)) } Strict Trs: { a__U11(X1, X2, X3) -> U11(X1, X2, X3) , a__U11(tt(), M, N) -> a__U12(tt(), M, N) , a__U12(X1, X2, X3) -> U12(X1, X2, X3) , a__U12(tt(), M, N) -> s(a__plus(mark(N), mark(M))) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, s(M)) -> a__U11(tt(), M, N) , a__plus(N, 0()) -> mark(N) , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) , mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3) , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) , a__U21(X1, X2, X3) -> U21(X1, X2, X3) , a__U21(tt(), M, N) -> a__U22(tt(), M, N) , a__U22(X1, X2, X3) -> U22(X1, X2, X3) , a__U22(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) , a__x(X1, X2) -> x(X1, X2) , a__x(N, s(M)) -> a__U21(tt(), M, N) , a__x(N, 0()) -> 0() } Weak DPs: { mark^#(tt()) -> c_8() , mark^#(0()) -> c_10() , a__x^#(N, 0()) -> c_23() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..