MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { U11(tt(), N) -> activate(N) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) , activate(n__isNat(X)) -> isNat(X) , activate(n__s(X)) -> s(activate(X)) , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) , s(X) -> n__s(X) , plus(X1, X2) -> n__plus(X1, X2) , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) , plus(N, 0()) -> U11(isNat(N), N) , U31(tt()) -> 0() , 0() -> n__0() , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) , x(N, 0()) -> U31(isNat(N)) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) } 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: { U11^#(tt(), N) -> c_1(activate^#(N)) , activate^#(X) -> c_2(X) , activate^#(n__0()) -> c_3(0^#()) , activate^#(n__plus(X1, X2)) -> c_4(plus^#(activate(X1), activate(X2))) , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) , activate^#(n__s(X)) -> c_6(s^#(activate(X))) , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) , 0^#() -> c_14() , plus^#(X1, X2) -> c_10(X1, X2) , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) , isNat^#(X) -> c_20(X) , isNat^#(n__0()) -> c_21() , isNat^#(n__plus(V1, V2)) -> c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) , isNat^#(n__x(V1, V2)) -> c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , s^#(X) -> c_9(X) , x^#(X1, X2) -> c_16(X1, X2) , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) , x^#(N, 0()) -> c_18(U31^#(isNat(N))) , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) , U31^#(tt()) -> c_13(0^#()) , U41^#(tt(), M, N) -> c_15(plus^#(x(activate(N), activate(M)), activate(N))) , and^#(tt(), X) -> c_19(activate^#(X)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { U11^#(tt(), N) -> c_1(activate^#(N)) , activate^#(X) -> c_2(X) , activate^#(n__0()) -> c_3(0^#()) , activate^#(n__plus(X1, X2)) -> c_4(plus^#(activate(X1), activate(X2))) , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) , activate^#(n__s(X)) -> c_6(s^#(activate(X))) , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) , 0^#() -> c_14() , plus^#(X1, X2) -> c_10(X1, X2) , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) , isNat^#(X) -> c_20(X) , isNat^#(n__0()) -> c_21() , isNat^#(n__plus(V1, V2)) -> c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) , isNat^#(n__x(V1, V2)) -> c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , s^#(X) -> c_9(X) , x^#(X1, X2) -> c_16(X1, X2) , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) , x^#(N, 0()) -> c_18(U31^#(isNat(N))) , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) , U31^#(tt()) -> c_13(0^#()) , U41^#(tt(), M, N) -> c_15(plus^#(x(activate(N), activate(M)), activate(N))) , and^#(tt(), X) -> c_19(activate^#(X)) } Strict Trs: { U11(tt(), N) -> activate(N) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) , activate(n__isNat(X)) -> isNat(X) , activate(n__s(X)) -> s(activate(X)) , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) , s(X) -> n__s(X) , plus(X1, X2) -> n__plus(X1, X2) , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) , plus(N, 0()) -> U11(isNat(N), N) , U31(tt()) -> 0() , 0() -> n__0() , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) , x(N, 0()) -> U31(isNat(N)) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {8,13} by applications of Pre({8,13}) = {2,3,5,9,12,15,17,18,22}. Here rules are labeled as follows: DPs: { 1: U11^#(tt(), N) -> c_1(activate^#(N)) , 2: activate^#(X) -> c_2(X) , 3: activate^#(n__0()) -> c_3(0^#()) , 4: activate^#(n__plus(X1, X2)) -> c_4(plus^#(activate(X1), activate(X2))) , 5: activate^#(n__isNat(X)) -> c_5(isNat^#(X)) , 6: activate^#(n__s(X)) -> c_6(s^#(activate(X))) , 7: activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) , 8: 0^#() -> c_14() , 9: plus^#(X1, X2) -> c_10(X1, X2) , 10: plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) , 11: plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) , 12: isNat^#(X) -> c_20(X) , 13: isNat^#(n__0()) -> c_21() , 14: isNat^#(n__plus(V1, V2)) -> c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , 15: isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) , 16: isNat^#(n__x(V1, V2)) -> c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , 17: s^#(X) -> c_9(X) , 18: x^#(X1, X2) -> c_16(X1, X2) , 19: x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) , 20: x^#(N, 0()) -> c_18(U31^#(isNat(N))) , 21: U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) , 22: U31^#(tt()) -> c_13(0^#()) , 23: U41^#(tt(), M, N) -> c_15(plus^#(x(activate(N), activate(M)), activate(N))) , 24: and^#(tt(), X) -> c_19(activate^#(X)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { U11^#(tt(), N) -> c_1(activate^#(N)) , activate^#(X) -> c_2(X) , activate^#(n__0()) -> c_3(0^#()) , activate^#(n__plus(X1, X2)) -> c_4(plus^#(activate(X1), activate(X2))) , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) , activate^#(n__s(X)) -> c_6(s^#(activate(X))) , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) , plus^#(X1, X2) -> c_10(X1, X2) , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) , isNat^#(X) -> c_20(X) , isNat^#(n__plus(V1, V2)) -> c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) , isNat^#(n__x(V1, V2)) -> c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , s^#(X) -> c_9(X) , x^#(X1, X2) -> c_16(X1, X2) , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) , x^#(N, 0()) -> c_18(U31^#(isNat(N))) , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) , U31^#(tt()) -> c_13(0^#()) , U41^#(tt(), M, N) -> c_15(plus^#(x(activate(N), activate(M)), activate(N))) , and^#(tt(), X) -> c_19(activate^#(X)) } Strict Trs: { U11(tt(), N) -> activate(N) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) , activate(n__isNat(X)) -> isNat(X) , activate(n__s(X)) -> s(activate(X)) , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) , s(X) -> n__s(X) , plus(X1, X2) -> n__plus(X1, X2) , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) , plus(N, 0()) -> U11(isNat(N), N) , U31(tt()) -> 0() , 0() -> n__0() , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) , x(N, 0()) -> U31(isNat(N)) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) } Weak DPs: { 0^#() -> c_14() , isNat^#(n__0()) -> c_21() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,20} by applications of Pre({3,20}) = {1,2,8,11,15,16,18,22}. Here rules are labeled as follows: DPs: { 1: U11^#(tt(), N) -> c_1(activate^#(N)) , 2: activate^#(X) -> c_2(X) , 3: activate^#(n__0()) -> c_3(0^#()) , 4: activate^#(n__plus(X1, X2)) -> c_4(plus^#(activate(X1), activate(X2))) , 5: activate^#(n__isNat(X)) -> c_5(isNat^#(X)) , 6: activate^#(n__s(X)) -> c_6(s^#(activate(X))) , 7: activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) , 8: plus^#(X1, X2) -> c_10(X1, X2) , 9: plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) , 10: plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) , 11: isNat^#(X) -> c_20(X) , 12: isNat^#(n__plus(V1, V2)) -> c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , 13: isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) , 14: isNat^#(n__x(V1, V2)) -> c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , 15: s^#(X) -> c_9(X) , 16: x^#(X1, X2) -> c_16(X1, X2) , 17: x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) , 18: x^#(N, 0()) -> c_18(U31^#(isNat(N))) , 19: U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) , 20: U31^#(tt()) -> c_13(0^#()) , 21: U41^#(tt(), M, N) -> c_15(plus^#(x(activate(N), activate(M)), activate(N))) , 22: and^#(tt(), X) -> c_19(activate^#(X)) , 23: 0^#() -> c_14() , 24: isNat^#(n__0()) -> c_21() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { U11^#(tt(), N) -> c_1(activate^#(N)) , activate^#(X) -> c_2(X) , activate^#(n__plus(X1, X2)) -> c_4(plus^#(activate(X1), activate(X2))) , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) , activate^#(n__s(X)) -> c_6(s^#(activate(X))) , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) , plus^#(X1, X2) -> c_10(X1, X2) , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) , isNat^#(X) -> c_20(X) , isNat^#(n__plus(V1, V2)) -> c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) , isNat^#(n__x(V1, V2)) -> c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , s^#(X) -> c_9(X) , x^#(X1, X2) -> c_16(X1, X2) , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) , x^#(N, 0()) -> c_18(U31^#(isNat(N))) , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) , U41^#(tt(), M, N) -> c_15(plus^#(x(activate(N), activate(M)), activate(N))) , and^#(tt(), X) -> c_19(activate^#(X)) } Strict Trs: { U11(tt(), N) -> activate(N) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) , activate(n__isNat(X)) -> isNat(X) , activate(n__s(X)) -> s(activate(X)) , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) , s(X) -> n__s(X) , plus(X1, X2) -> n__plus(X1, X2) , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) , plus(N, 0()) -> U11(isNat(N), N) , U31(tt()) -> 0() , 0() -> n__0() , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) , x(N, 0()) -> U31(isNat(N)) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) } Weak DPs: { activate^#(n__0()) -> c_3(0^#()) , 0^#() -> c_14() , isNat^#(n__0()) -> c_21() , U31^#(tt()) -> c_13(0^#()) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {17} by applications of Pre({17}) = {2,6,7,10,14,15}. Here rules are labeled as follows: DPs: { 1: U11^#(tt(), N) -> c_1(activate^#(N)) , 2: activate^#(X) -> c_2(X) , 3: activate^#(n__plus(X1, X2)) -> c_4(plus^#(activate(X1), activate(X2))) , 4: activate^#(n__isNat(X)) -> c_5(isNat^#(X)) , 5: activate^#(n__s(X)) -> c_6(s^#(activate(X))) , 6: activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) , 7: plus^#(X1, X2) -> c_10(X1, X2) , 8: plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) , 9: plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) , 10: isNat^#(X) -> c_20(X) , 11: isNat^#(n__plus(V1, V2)) -> c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , 12: isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) , 13: isNat^#(n__x(V1, V2)) -> c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , 14: s^#(X) -> c_9(X) , 15: x^#(X1, X2) -> c_16(X1, X2) , 16: x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) , 17: x^#(N, 0()) -> c_18(U31^#(isNat(N))) , 18: U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) , 19: U41^#(tt(), M, N) -> c_15(plus^#(x(activate(N), activate(M)), activate(N))) , 20: and^#(tt(), X) -> c_19(activate^#(X)) , 21: activate^#(n__0()) -> c_3(0^#()) , 22: 0^#() -> c_14() , 23: isNat^#(n__0()) -> c_21() , 24: U31^#(tt()) -> c_13(0^#()) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { U11^#(tt(), N) -> c_1(activate^#(N)) , activate^#(X) -> c_2(X) , activate^#(n__plus(X1, X2)) -> c_4(plus^#(activate(X1), activate(X2))) , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) , activate^#(n__s(X)) -> c_6(s^#(activate(X))) , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) , plus^#(X1, X2) -> c_10(X1, X2) , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) , isNat^#(X) -> c_20(X) , isNat^#(n__plus(V1, V2)) -> c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) , isNat^#(n__x(V1, V2)) -> c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , s^#(X) -> c_9(X) , x^#(X1, X2) -> c_16(X1, X2) , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) , U41^#(tt(), M, N) -> c_15(plus^#(x(activate(N), activate(M)), activate(N))) , and^#(tt(), X) -> c_19(activate^#(X)) } Strict Trs: { U11(tt(), N) -> activate(N) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) , activate(n__isNat(X)) -> isNat(X) , activate(n__s(X)) -> s(activate(X)) , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) , s(X) -> n__s(X) , plus(X1, X2) -> n__plus(X1, X2) , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) , plus(N, 0()) -> U11(isNat(N), N) , U31(tt()) -> 0() , 0() -> n__0() , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) , x(N, 0()) -> U31(isNat(N)) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) } Weak DPs: { activate^#(n__0()) -> c_3(0^#()) , 0^#() -> c_14() , isNat^#(n__0()) -> c_21() , x^#(N, 0()) -> c_18(U31^#(isNat(N))) , U31^#(tt()) -> c_13(0^#()) } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..