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)) , 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) , 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)) , 0() -> n__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: { 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))) , 0^#() -> c_17() , plus^#(X1, X2) -> c_9(X1, X2) , plus^#(N, s(M)) -> c_10(U21^#(and(isNat(M), n__isNat(N)), M, N)) , plus^#(N, 0()) -> c_11(U11^#(isNat(N), N)) , isNat^#(X) -> c_13(X) , isNat^#(n__0()) -> c_14() , isNat^#(n__plus(V1, V2)) -> c_15(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , isNat^#(n__s(V1)) -> c_16(isNat^#(activate(V1))) , s^#(X) -> c_8(X) , U21^#(tt(), M, N) -> c_7(s^#(plus(activate(N), activate(M)))) , and^#(tt(), X) -> c_12(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))) , 0^#() -> c_17() , plus^#(X1, X2) -> c_9(X1, X2) , plus^#(N, s(M)) -> c_10(U21^#(and(isNat(M), n__isNat(N)), M, N)) , plus^#(N, 0()) -> c_11(U11^#(isNat(N), N)) , isNat^#(X) -> c_13(X) , isNat^#(n__0()) -> c_14() , isNat^#(n__plus(V1, V2)) -> c_15(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , isNat^#(n__s(V1)) -> c_16(isNat^#(activate(V1))) , s^#(X) -> c_8(X) , U21^#(tt(), M, N) -> c_7(s^#(plus(activate(N), activate(M)))) , and^#(tt(), X) -> c_12(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)) , 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) , 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)) , 0() -> n__0() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {7,12} by applications of Pre({7,12}) = {2,3,5,8,11,14,15}. Here rules are labeled as follows: DPs: { 1: U11^#(tt(), N) -> c_1(activate^#(N)) , 2: activate^#(X) -> c_2(X) , 3: activate^#(n__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: 0^#() -> c_17() , 8: plus^#(X1, X2) -> c_9(X1, X2) , 9: plus^#(N, s(M)) -> c_10(U21^#(and(isNat(M), n__isNat(N)), M, N)) , 10: plus^#(N, 0()) -> c_11(U11^#(isNat(N), N)) , 11: isNat^#(X) -> c_13(X) , 12: isNat^#(n__0()) -> c_14() , 13: isNat^#(n__plus(V1, V2)) -> c_15(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , 14: isNat^#(n__s(V1)) -> c_16(isNat^#(activate(V1))) , 15: s^#(X) -> c_8(X) , 16: U21^#(tt(), M, N) -> c_7(s^#(plus(activate(N), activate(M)))) , 17: and^#(tt(), X) -> c_12(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))) , plus^#(X1, X2) -> c_9(X1, X2) , plus^#(N, s(M)) -> c_10(U21^#(and(isNat(M), n__isNat(N)), M, N)) , plus^#(N, 0()) -> c_11(U11^#(isNat(N), N)) , isNat^#(X) -> c_13(X) , isNat^#(n__plus(V1, V2)) -> c_15(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , isNat^#(n__s(V1)) -> c_16(isNat^#(activate(V1))) , s^#(X) -> c_8(X) , U21^#(tt(), M, N) -> c_7(s^#(plus(activate(N), activate(M)))) , and^#(tt(), X) -> c_12(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)) , 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) , 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)) , 0() -> n__0() } Weak DPs: { 0^#() -> c_17() , isNat^#(n__0()) -> c_14() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3} by applications of Pre({3}) = {1,2,7,10,13,15}. Here rules are labeled as follows: DPs: { 1: U11^#(tt(), N) -> c_1(activate^#(N)) , 2: activate^#(X) -> c_2(X) , 3: activate^#(n__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: plus^#(X1, X2) -> c_9(X1, X2) , 8: plus^#(N, s(M)) -> c_10(U21^#(and(isNat(M), n__isNat(N)), M, N)) , 9: plus^#(N, 0()) -> c_11(U11^#(isNat(N), N)) , 10: isNat^#(X) -> c_13(X) , 11: isNat^#(n__plus(V1, V2)) -> c_15(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , 12: isNat^#(n__s(V1)) -> c_16(isNat^#(activate(V1))) , 13: s^#(X) -> c_8(X) , 14: U21^#(tt(), M, N) -> c_7(s^#(plus(activate(N), activate(M)))) , 15: and^#(tt(), X) -> c_12(activate^#(X)) , 16: 0^#() -> c_17() , 17: isNat^#(n__0()) -> c_14() } 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))) , plus^#(X1, X2) -> c_9(X1, X2) , plus^#(N, s(M)) -> c_10(U21^#(and(isNat(M), n__isNat(N)), M, N)) , plus^#(N, 0()) -> c_11(U11^#(isNat(N), N)) , isNat^#(X) -> c_13(X) , isNat^#(n__plus(V1, V2)) -> c_15(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) , isNat^#(n__s(V1)) -> c_16(isNat^#(activate(V1))) , s^#(X) -> c_8(X) , U21^#(tt(), M, N) -> c_7(s^#(plus(activate(N), activate(M)))) , and^#(tt(), X) -> c_12(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)) , 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) , 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)) , 0() -> n__0() } Weak DPs: { activate^#(n__0()) -> c_3(0^#()) , 0^#() -> c_17() , isNat^#(n__0()) -> c_14() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..