MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { U11(tt(), V2) -> U12(isNat(activate(V2))) , U12(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) , isNat(n__s(V1)) -> U21(isNat(activate(V1))) , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(X1, X2) , activate(n__s(X)) -> s(X) , activate(n__x(X1, X2)) -> x(X1, X2) , U21(tt()) -> tt() , U31(tt(), V2) -> U32(isNat(activate(V2))) , U32(tt()) -> tt() , U41(tt(), N) -> activate(N) , U51(tt(), M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) , U52(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)) -> U51(isNat(M), M, N) , plus(N, 0()) -> U41(isNat(N), N) , U61(tt()) -> 0() , 0() -> n__0() , U71(tt(), M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U71(isNat(M), M, N) , x(N, 0()) -> U61(isNat(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) '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 minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-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(), V2) -> c_1(U12^#(isNat(activate(V2)))) , U12^#(tt()) -> c_2() , isNat^#(n__0()) -> c_3() , isNat^#(n__plus(V1, V2)) -> c_4(U11^#(isNat(activate(V1)), activate(V2))) , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) , isNat^#(n__x(V1, V2)) -> c_6(U31^#(isNat(activate(V1)), activate(V2))) , U21^#(tt()) -> c_12() , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) , activate^#(X) -> c_7(X) , activate^#(n__0()) -> c_8(0^#()) , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) , activate^#(n__s(X)) -> c_10(s^#(X)) , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) , 0^#() -> c_23() , plus^#(X1, X2) -> c_19(X1, X2) , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) , s^#(X) -> c_18(X) , x^#(X1, X2) -> c_26(X1, X2) , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) , x^#(N, 0()) -> c_28(U61^#(isNat(N))) , U32^#(tt()) -> c_14() , U41^#(tt(), N) -> c_15(activate^#(N)) , U51^#(tt(), M, N) -> c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) , U61^#(tt()) -> c_22(0^#()) , U71^#(tt(), M, N) -> c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) , U72^#(tt(), M, N) -> c_25(plus^#(x(activate(N), activate(M)), activate(N))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) , U12^#(tt()) -> c_2() , isNat^#(n__0()) -> c_3() , isNat^#(n__plus(V1, V2)) -> c_4(U11^#(isNat(activate(V1)), activate(V2))) , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) , isNat^#(n__x(V1, V2)) -> c_6(U31^#(isNat(activate(V1)), activate(V2))) , U21^#(tt()) -> c_12() , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) , activate^#(X) -> c_7(X) , activate^#(n__0()) -> c_8(0^#()) , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) , activate^#(n__s(X)) -> c_10(s^#(X)) , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) , 0^#() -> c_23() , plus^#(X1, X2) -> c_19(X1, X2) , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) , s^#(X) -> c_18(X) , x^#(X1, X2) -> c_26(X1, X2) , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) , x^#(N, 0()) -> c_28(U61^#(isNat(N))) , U32^#(tt()) -> c_14() , U41^#(tt(), N) -> c_15(activate^#(N)) , U51^#(tt(), M, N) -> c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) , U61^#(tt()) -> c_22(0^#()) , U71^#(tt(), M, N) -> c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) , U72^#(tt(), M, N) -> c_25(plus^#(x(activate(N), activate(M)), activate(N))) } Strict Trs: { U11(tt(), V2) -> U12(isNat(activate(V2))) , U12(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) , isNat(n__s(V1)) -> U21(isNat(activate(V1))) , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(X1, X2) , activate(n__s(X)) -> s(X) , activate(n__x(X1, X2)) -> x(X1, X2) , U21(tt()) -> tt() , U31(tt(), V2) -> U32(isNat(activate(V2))) , U32(tt()) -> tt() , U41(tt(), N) -> activate(N) , U51(tt(), M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) , U52(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)) -> U51(isNat(M), M, N) , plus(N, 0()) -> U41(isNat(N), N) , U61(tt()) -> 0() , 0() -> n__0() , U71(tt(), M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U71(isNat(M), M, N) , x(N, 0()) -> U61(isNat(N)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,3,7,14,22} by applications of Pre({2,3,7,14,22}) = {1,5,8,9,10,15,18,19,26}. Here rules are labeled as follows: DPs: { 1: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) , 2: U12^#(tt()) -> c_2() , 3: isNat^#(n__0()) -> c_3() , 4: isNat^#(n__plus(V1, V2)) -> c_4(U11^#(isNat(activate(V1)), activate(V2))) , 5: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) , 6: isNat^#(n__x(V1, V2)) -> c_6(U31^#(isNat(activate(V1)), activate(V2))) , 7: U21^#(tt()) -> c_12() , 8: U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) , 9: activate^#(X) -> c_7(X) , 10: activate^#(n__0()) -> c_8(0^#()) , 11: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) , 12: activate^#(n__s(X)) -> c_10(s^#(X)) , 13: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) , 14: 0^#() -> c_23() , 15: plus^#(X1, X2) -> c_19(X1, X2) , 16: plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) , 17: plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) , 18: s^#(X) -> c_18(X) , 19: x^#(X1, X2) -> c_26(X1, X2) , 20: x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) , 21: x^#(N, 0()) -> c_28(U61^#(isNat(N))) , 22: U32^#(tt()) -> c_14() , 23: U41^#(tt(), N) -> c_15(activate^#(N)) , 24: U51^#(tt(), M, N) -> c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) , 25: U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) , 26: U61^#(tt()) -> c_22(0^#()) , 27: U71^#(tt(), M, N) -> c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) , 28: U72^#(tt(), M, N) -> c_25(plus^#(x(activate(N), activate(M)), activate(N))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) , isNat^#(n__plus(V1, V2)) -> c_4(U11^#(isNat(activate(V1)), activate(V2))) , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) , isNat^#(n__x(V1, V2)) -> c_6(U31^#(isNat(activate(V1)), activate(V2))) , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) , activate^#(X) -> c_7(X) , activate^#(n__0()) -> c_8(0^#()) , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) , activate^#(n__s(X)) -> c_10(s^#(X)) , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) , plus^#(X1, X2) -> c_19(X1, X2) , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) , s^#(X) -> c_18(X) , x^#(X1, X2) -> c_26(X1, X2) , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) , x^#(N, 0()) -> c_28(U61^#(isNat(N))) , U41^#(tt(), N) -> c_15(activate^#(N)) , U51^#(tt(), M, N) -> c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) , U61^#(tt()) -> c_22(0^#()) , U71^#(tt(), M, N) -> c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) , U72^#(tt(), M, N) -> c_25(plus^#(x(activate(N), activate(M)), activate(N))) } Strict Trs: { U11(tt(), V2) -> U12(isNat(activate(V2))) , U12(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) , isNat(n__s(V1)) -> U21(isNat(activate(V1))) , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(X1, X2) , activate(n__s(X)) -> s(X) , activate(n__x(X1, X2)) -> x(X1, X2) , U21(tt()) -> tt() , U31(tt(), V2) -> U32(isNat(activate(V2))) , U32(tt()) -> tt() , U41(tt(), N) -> activate(N) , U51(tt(), M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) , U52(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)) -> U51(isNat(M), M, N) , plus(N, 0()) -> U41(isNat(N), N) , U61(tt()) -> 0() , 0() -> n__0() , U71(tt(), M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U71(isNat(M), M, N) , x(N, 0()) -> U61(isNat(N)) } Weak DPs: { U12^#(tt()) -> c_2() , isNat^#(n__0()) -> c_3() , U21^#(tt()) -> c_12() , 0^#() -> c_23() , U32^#(tt()) -> c_14() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,3,5,7,21} by applications of Pre({1,3,5,7,21}) = {2,4,6,11,14,15,17,18}. Here rules are labeled as follows: DPs: { 1: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) , 2: isNat^#(n__plus(V1, V2)) -> c_4(U11^#(isNat(activate(V1)), activate(V2))) , 3: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) , 4: isNat^#(n__x(V1, V2)) -> c_6(U31^#(isNat(activate(V1)), activate(V2))) , 5: U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) , 6: activate^#(X) -> c_7(X) , 7: activate^#(n__0()) -> c_8(0^#()) , 8: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) , 9: activate^#(n__s(X)) -> c_10(s^#(X)) , 10: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) , 11: plus^#(X1, X2) -> c_19(X1, X2) , 12: plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) , 13: plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) , 14: s^#(X) -> c_18(X) , 15: x^#(X1, X2) -> c_26(X1, X2) , 16: x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) , 17: x^#(N, 0()) -> c_28(U61^#(isNat(N))) , 18: U41^#(tt(), N) -> c_15(activate^#(N)) , 19: U51^#(tt(), M, N) -> c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) , 20: U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) , 21: U61^#(tt()) -> c_22(0^#()) , 22: U71^#(tt(), M, N) -> c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) , 23: U72^#(tt(), M, N) -> c_25(plus^#(x(activate(N), activate(M)), activate(N))) , 24: U12^#(tt()) -> c_2() , 25: isNat^#(n__0()) -> c_3() , 26: U21^#(tt()) -> c_12() , 27: 0^#() -> c_23() , 28: U32^#(tt()) -> c_14() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { isNat^#(n__plus(V1, V2)) -> c_4(U11^#(isNat(activate(V1)), activate(V2))) , isNat^#(n__x(V1, V2)) -> c_6(U31^#(isNat(activate(V1)), activate(V2))) , activate^#(X) -> c_7(X) , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) , activate^#(n__s(X)) -> c_10(s^#(X)) , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) , plus^#(X1, X2) -> c_19(X1, X2) , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) , s^#(X) -> c_18(X) , x^#(X1, X2) -> c_26(X1, X2) , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) , x^#(N, 0()) -> c_28(U61^#(isNat(N))) , U41^#(tt(), N) -> c_15(activate^#(N)) , U51^#(tt(), M, N) -> c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) , U71^#(tt(), M, N) -> c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) , U72^#(tt(), M, N) -> c_25(plus^#(x(activate(N), activate(M)), activate(N))) } Strict Trs: { U11(tt(), V2) -> U12(isNat(activate(V2))) , U12(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) , isNat(n__s(V1)) -> U21(isNat(activate(V1))) , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(X1, X2) , activate(n__s(X)) -> s(X) , activate(n__x(X1, X2)) -> x(X1, X2) , U21(tt()) -> tt() , U31(tt(), V2) -> U32(isNat(activate(V2))) , U32(tt()) -> tt() , U41(tt(), N) -> activate(N) , U51(tt(), M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) , U52(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)) -> U51(isNat(M), M, N) , plus(N, 0()) -> U41(isNat(N), N) , U61(tt()) -> 0() , 0() -> n__0() , U71(tt(), M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U71(isNat(M), M, N) , x(N, 0()) -> U61(isNat(N)) } Weak DPs: { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) , U12^#(tt()) -> c_2() , isNat^#(n__0()) -> c_3() , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) , U21^#(tt()) -> c_12() , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) , activate^#(n__0()) -> c_8(0^#()) , 0^#() -> c_23() , U32^#(tt()) -> c_14() , U61^#(tt()) -> c_22(0^#()) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,13} by applications of Pre({1,2,13}) = {3,6,7,10,11}. Here rules are labeled as follows: DPs: { 1: isNat^#(n__plus(V1, V2)) -> c_4(U11^#(isNat(activate(V1)), activate(V2))) , 2: isNat^#(n__x(V1, V2)) -> c_6(U31^#(isNat(activate(V1)), activate(V2))) , 3: activate^#(X) -> c_7(X) , 4: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) , 5: activate^#(n__s(X)) -> c_10(s^#(X)) , 6: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) , 7: plus^#(X1, X2) -> c_19(X1, X2) , 8: plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) , 9: plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) , 10: s^#(X) -> c_18(X) , 11: x^#(X1, X2) -> c_26(X1, X2) , 12: x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) , 13: x^#(N, 0()) -> c_28(U61^#(isNat(N))) , 14: U41^#(tt(), N) -> c_15(activate^#(N)) , 15: U51^#(tt(), M, N) -> c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) , 16: U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) , 17: U71^#(tt(), M, N) -> c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) , 18: U72^#(tt(), M, N) -> c_25(plus^#(x(activate(N), activate(M)), activate(N))) , 19: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) , 20: U12^#(tt()) -> c_2() , 21: isNat^#(n__0()) -> c_3() , 22: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) , 23: U21^#(tt()) -> c_12() , 24: U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) , 25: activate^#(n__0()) -> c_8(0^#()) , 26: 0^#() -> c_23() , 27: U32^#(tt()) -> c_14() , 28: U61^#(tt()) -> c_22(0^#()) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { activate^#(X) -> c_7(X) , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) , activate^#(n__s(X)) -> c_10(s^#(X)) , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) , plus^#(X1, X2) -> c_19(X1, X2) , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) , s^#(X) -> c_18(X) , x^#(X1, X2) -> c_26(X1, X2) , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) , U41^#(tt(), N) -> c_15(activate^#(N)) , U51^#(tt(), M, N) -> c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) , U71^#(tt(), M, N) -> c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) , U72^#(tt(), M, N) -> c_25(plus^#(x(activate(N), activate(M)), activate(N))) } Strict Trs: { U11(tt(), V2) -> U12(isNat(activate(V2))) , U12(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) , isNat(n__s(V1)) -> U21(isNat(activate(V1))) , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) , activate(X) -> X , activate(n__0()) -> 0() , activate(n__plus(X1, X2)) -> plus(X1, X2) , activate(n__s(X)) -> s(X) , activate(n__x(X1, X2)) -> x(X1, X2) , U21(tt()) -> tt() , U31(tt(), V2) -> U32(isNat(activate(V2))) , U32(tt()) -> tt() , U41(tt(), N) -> activate(N) , U51(tt(), M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) , U52(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)) -> U51(isNat(M), M, N) , plus(N, 0()) -> U41(isNat(N), N) , U61(tt()) -> 0() , 0() -> n__0() , U71(tt(), M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) , x(X1, X2) -> n__x(X1, X2) , x(N, s(M)) -> U71(isNat(M), M, N) , x(N, 0()) -> U61(isNat(N)) } Weak DPs: { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) , U12^#(tt()) -> c_2() , isNat^#(n__0()) -> c_3() , isNat^#(n__plus(V1, V2)) -> c_4(U11^#(isNat(activate(V1)), activate(V2))) , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) , isNat^#(n__x(V1, V2)) -> c_6(U31^#(isNat(activate(V1)), activate(V2))) , U21^#(tt()) -> c_12() , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) , activate^#(n__0()) -> c_8(0^#()) , 0^#() -> c_23() , x^#(N, 0()) -> c_28(U61^#(isNat(N))) , U32^#(tt()) -> c_14() , U61^#(tt()) -> c_22(0^#()) } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..