MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__U11(X1, X2) -> U11(X1, X2) , a__U11(tt(), V2) -> a__U12(a__isNat(V2)) , a__U12(X) -> U12(X) , a__U12(tt()) -> tt() , a__isNat(X) -> isNat(X) , a__isNat(s(V1)) -> a__U21(a__isNat(V1)) , a__isNat(0()) -> tt() , a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) , a__U21(X) -> U21(X) , a__U21(tt()) -> tt() , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), N) -> mark(N) , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) , mark(U12(X)) -> a__U12(mark(X)) , mark(isNat(X)) -> a__isNat(X) , mark(U21(X)) -> a__U21(mark(X)) , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) , mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) , a__U41(X1, X2, X3) -> U41(X1, X2, X3) , a__U41(tt(), M, N) -> a__U42(a__isNat(N), M, N) , a__U42(X1, X2, X3) -> U42(X1, X2, X3) , a__U42(tt(), M, N) -> s(a__plus(mark(N), mark(M))) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) , a__plus(N, 0()) -> a__U31(a__isNat(N), N) } 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) '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) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) '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 2) '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 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) -> c_1(X1, X2) , a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2))) , a__U12^#(X) -> c_3(X) , a__U12^#(tt()) -> c_4() , a__isNat^#(X) -> c_5(X) , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1))) , a__isNat^#(0()) -> c_7() , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2)) , a__U21^#(X) -> c_9(X) , a__U21^#(tt()) -> c_10() , a__U31^#(X1, X2) -> c_11(X1, X2) , a__U31^#(tt(), N) -> c_12(mark^#(N)) , mark^#(tt()) -> c_13() , mark^#(s(X)) -> c_14(mark^#(X)) , mark^#(0()) -> c_15() , mark^#(plus(X1, X2)) -> c_16(a__plus^#(mark(X1), mark(X2))) , mark^#(U11(X1, X2)) -> c_17(a__U11^#(mark(X1), X2)) , mark^#(U12(X)) -> c_18(a__U12^#(mark(X))) , mark^#(isNat(X)) -> c_19(a__isNat^#(X)) , mark^#(U21(X)) -> c_20(a__U21^#(mark(X))) , mark^#(U31(X1, X2)) -> c_21(a__U31^#(mark(X1), X2)) , mark^#(U41(X1, X2, X3)) -> c_22(a__U41^#(mark(X1), X2, X3)) , mark^#(U42(X1, X2, X3)) -> c_23(a__U42^#(mark(X1), X2, X3)) , a__plus^#(X1, X2) -> c_28(X1, X2) , a__plus^#(N, s(M)) -> c_29(a__U41^#(a__isNat(M), M, N)) , a__plus^#(N, 0()) -> c_30(a__U31^#(a__isNat(N), N)) , a__U41^#(X1, X2, X3) -> c_24(X1, X2, X3) , a__U41^#(tt(), M, N) -> c_25(a__U42^#(a__isNat(N), M, N)) , a__U42^#(X1, X2, X3) -> c_26(X1, X2, X3) , a__U42^#(tt(), M, N) -> c_27(a__plus^#(mark(N), mark(M))) } 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) -> c_1(X1, X2) , a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2))) , a__U12^#(X) -> c_3(X) , a__U12^#(tt()) -> c_4() , a__isNat^#(X) -> c_5(X) , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1))) , a__isNat^#(0()) -> c_7() , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2)) , a__U21^#(X) -> c_9(X) , a__U21^#(tt()) -> c_10() , a__U31^#(X1, X2) -> c_11(X1, X2) , a__U31^#(tt(), N) -> c_12(mark^#(N)) , mark^#(tt()) -> c_13() , mark^#(s(X)) -> c_14(mark^#(X)) , mark^#(0()) -> c_15() , mark^#(plus(X1, X2)) -> c_16(a__plus^#(mark(X1), mark(X2))) , mark^#(U11(X1, X2)) -> c_17(a__U11^#(mark(X1), X2)) , mark^#(U12(X)) -> c_18(a__U12^#(mark(X))) , mark^#(isNat(X)) -> c_19(a__isNat^#(X)) , mark^#(U21(X)) -> c_20(a__U21^#(mark(X))) , mark^#(U31(X1, X2)) -> c_21(a__U31^#(mark(X1), X2)) , mark^#(U41(X1, X2, X3)) -> c_22(a__U41^#(mark(X1), X2, X3)) , mark^#(U42(X1, X2, X3)) -> c_23(a__U42^#(mark(X1), X2, X3)) , a__plus^#(X1, X2) -> c_28(X1, X2) , a__plus^#(N, s(M)) -> c_29(a__U41^#(a__isNat(M), M, N)) , a__plus^#(N, 0()) -> c_30(a__U31^#(a__isNat(N), N)) , a__U41^#(X1, X2, X3) -> c_24(X1, X2, X3) , a__U41^#(tt(), M, N) -> c_25(a__U42^#(a__isNat(N), M, N)) , a__U42^#(X1, X2, X3) -> c_26(X1, X2, X3) , a__U42^#(tt(), M, N) -> c_27(a__plus^#(mark(N), mark(M))) } Strict Trs: { a__U11(X1, X2) -> U11(X1, X2) , a__U11(tt(), V2) -> a__U12(a__isNat(V2)) , a__U12(X) -> U12(X) , a__U12(tt()) -> tt() , a__isNat(X) -> isNat(X) , a__isNat(s(V1)) -> a__U21(a__isNat(V1)) , a__isNat(0()) -> tt() , a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) , a__U21(X) -> U21(X) , a__U21(tt()) -> tt() , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), N) -> mark(N) , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) , mark(U12(X)) -> a__U12(mark(X)) , mark(isNat(X)) -> a__isNat(X) , mark(U21(X)) -> a__U21(mark(X)) , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) , mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) , a__U41(X1, X2, X3) -> U41(X1, X2, X3) , a__U41(tt(), M, N) -> a__U42(a__isNat(N), M, N) , a__U42(X1, X2, X3) -> U42(X1, X2, X3) , a__U42(tt(), M, N) -> s(a__plus(mark(N), mark(M))) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) , a__plus(N, 0()) -> a__U31(a__isNat(N), N) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,7,10,13,15} by applications of Pre({4,7,10,13,15}) = {1,2,3,5,6,9,11,12,14,18,19,20,24,27,29}. Here rules are labeled as follows: DPs: { 1: a__U11^#(X1, X2) -> c_1(X1, X2) , 2: a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2))) , 3: a__U12^#(X) -> c_3(X) , 4: a__U12^#(tt()) -> c_4() , 5: a__isNat^#(X) -> c_5(X) , 6: a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1))) , 7: a__isNat^#(0()) -> c_7() , 8: a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2)) , 9: a__U21^#(X) -> c_9(X) , 10: a__U21^#(tt()) -> c_10() , 11: a__U31^#(X1, X2) -> c_11(X1, X2) , 12: a__U31^#(tt(), N) -> c_12(mark^#(N)) , 13: mark^#(tt()) -> c_13() , 14: mark^#(s(X)) -> c_14(mark^#(X)) , 15: mark^#(0()) -> c_15() , 16: mark^#(plus(X1, X2)) -> c_16(a__plus^#(mark(X1), mark(X2))) , 17: mark^#(U11(X1, X2)) -> c_17(a__U11^#(mark(X1), X2)) , 18: mark^#(U12(X)) -> c_18(a__U12^#(mark(X))) , 19: mark^#(isNat(X)) -> c_19(a__isNat^#(X)) , 20: mark^#(U21(X)) -> c_20(a__U21^#(mark(X))) , 21: mark^#(U31(X1, X2)) -> c_21(a__U31^#(mark(X1), X2)) , 22: mark^#(U41(X1, X2, X3)) -> c_22(a__U41^#(mark(X1), X2, X3)) , 23: mark^#(U42(X1, X2, X3)) -> c_23(a__U42^#(mark(X1), X2, X3)) , 24: a__plus^#(X1, X2) -> c_28(X1, X2) , 25: a__plus^#(N, s(M)) -> c_29(a__U41^#(a__isNat(M), M, N)) , 26: a__plus^#(N, 0()) -> c_30(a__U31^#(a__isNat(N), N)) , 27: a__U41^#(X1, X2, X3) -> c_24(X1, X2, X3) , 28: a__U41^#(tt(), M, N) -> c_25(a__U42^#(a__isNat(N), M, N)) , 29: a__U42^#(X1, X2, X3) -> c_26(X1, X2, X3) , 30: a__U42^#(tt(), M, N) -> c_27(a__plus^#(mark(N), mark(M))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(X1, X2) -> c_1(X1, X2) , a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2))) , a__U12^#(X) -> c_3(X) , a__isNat^#(X) -> c_5(X) , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1))) , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2)) , a__U21^#(X) -> c_9(X) , a__U31^#(X1, X2) -> c_11(X1, X2) , a__U31^#(tt(), N) -> c_12(mark^#(N)) , mark^#(s(X)) -> c_14(mark^#(X)) , mark^#(plus(X1, X2)) -> c_16(a__plus^#(mark(X1), mark(X2))) , mark^#(U11(X1, X2)) -> c_17(a__U11^#(mark(X1), X2)) , mark^#(U12(X)) -> c_18(a__U12^#(mark(X))) , mark^#(isNat(X)) -> c_19(a__isNat^#(X)) , mark^#(U21(X)) -> c_20(a__U21^#(mark(X))) , mark^#(U31(X1, X2)) -> c_21(a__U31^#(mark(X1), X2)) , mark^#(U41(X1, X2, X3)) -> c_22(a__U41^#(mark(X1), X2, X3)) , mark^#(U42(X1, X2, X3)) -> c_23(a__U42^#(mark(X1), X2, X3)) , a__plus^#(X1, X2) -> c_28(X1, X2) , a__plus^#(N, s(M)) -> c_29(a__U41^#(a__isNat(M), M, N)) , a__plus^#(N, 0()) -> c_30(a__U31^#(a__isNat(N), N)) , a__U41^#(X1, X2, X3) -> c_24(X1, X2, X3) , a__U41^#(tt(), M, N) -> c_25(a__U42^#(a__isNat(N), M, N)) , a__U42^#(X1, X2, X3) -> c_26(X1, X2, X3) , a__U42^#(tt(), M, N) -> c_27(a__plus^#(mark(N), mark(M))) } Strict Trs: { a__U11(X1, X2) -> U11(X1, X2) , a__U11(tt(), V2) -> a__U12(a__isNat(V2)) , a__U12(X) -> U12(X) , a__U12(tt()) -> tt() , a__isNat(X) -> isNat(X) , a__isNat(s(V1)) -> a__U21(a__isNat(V1)) , a__isNat(0()) -> tt() , a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) , a__U21(X) -> U21(X) , a__U21(tt()) -> tt() , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), N) -> mark(N) , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) , mark(U12(X)) -> a__U12(mark(X)) , mark(isNat(X)) -> a__isNat(X) , mark(U21(X)) -> a__U21(mark(X)) , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) , mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) , a__U41(X1, X2, X3) -> U41(X1, X2, X3) , a__U41(tt(), M, N) -> a__U42(a__isNat(N), M, N) , a__U42(X1, X2, X3) -> U42(X1, X2, X3) , a__U42(tt(), M, N) -> s(a__plus(mark(N), mark(M))) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) , a__plus(N, 0()) -> a__U31(a__isNat(N), N) } Weak DPs: { a__U12^#(tt()) -> c_4() , a__isNat^#(0()) -> c_7() , a__U21^#(tt()) -> c_10() , mark^#(tt()) -> c_13() , mark^#(0()) -> c_15() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..