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__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) , a__U21(X) -> U21(X) , a__U21(tt()) -> tt() , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), V2) -> a__U32(a__isNat(V2)) , a__U32(X) -> U32(X) , a__U32(tt()) -> tt() , a__U41(X1, X2) -> U41(X1, X2) , a__U41(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(x(X1, X2)) -> a__x(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(U32(X)) -> a__U32(mark(X)) , mark(U41(X1, X2)) -> a__U41(mark(X1), X2) , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) , mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) , mark(U61(X)) -> a__U61(mark(X)) , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) , mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) , a__U51(X1, X2, X3) -> U51(X1, X2, X3) , a__U51(tt(), M, N) -> a__U52(a__isNat(N), M, N) , a__U52(X1, X2, X3) -> U52(X1, X2, X3) , a__U52(tt(), M, N) -> s(a__plus(mark(N), mark(M))) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) , a__plus(N, 0()) -> a__U41(a__isNat(N), N) , a__U61(X) -> U61(X) , a__U61(tt()) -> 0() , a__U71(X1, X2, X3) -> U71(X1, X2, X3) , a__U71(tt(), M, N) -> a__U72(a__isNat(N), M, N) , a__U72(X1, X2, X3) -> U72(X1, X2, X3) , a__U72(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__U71(a__isNat(M), M, N) , a__x(N, 0()) -> a__U61(a__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) '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) '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) '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__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2)) , a__U21^#(X) -> c_10(X) , a__U21^#(tt()) -> c_11() , a__U31^#(X1, X2) -> c_12(X1, X2) , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2))) , a__U32^#(X) -> c_14(X) , a__U32^#(tt()) -> c_15() , a__U41^#(X1, X2) -> c_16(X1, X2) , a__U41^#(tt(), N) -> c_17(mark^#(N)) , mark^#(tt()) -> c_18() , mark^#(s(X)) -> c_19(mark^#(X)) , mark^#(0()) -> c_20() , mark^#(plus(X1, X2)) -> c_21(a__plus^#(mark(X1), mark(X2))) , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2))) , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2)) , mark^#(U12(X)) -> c_24(a__U12^#(mark(X))) , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) , mark^#(U21(X)) -> c_26(a__U21^#(mark(X))) , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2)) , mark^#(U32(X)) -> c_28(a__U32^#(mark(X))) , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2)) , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3)) , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3)) , mark^#(U61(X)) -> c_32(a__U61^#(mark(X))) , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3)) , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3)) , a__plus^#(X1, X2) -> c_39(X1, X2) , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N)) , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N)) , a__x^#(X1, X2) -> c_48(X1, X2) , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N)) , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N))) , a__U51^#(X1, X2, X3) -> c_35(X1, X2, X3) , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N)) , a__U52^#(X1, X2, X3) -> c_37(X1, X2, X3) , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M))) , a__U61^#(X) -> c_42(X) , a__U61^#(tt()) -> c_43() , a__U71^#(X1, X2, X3) -> c_44(X1, X2, X3) , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N)) , a__U72^#(X1, X2, X3) -> c_46(X1, X2, X3) , a__U72^#(tt(), M, N) -> c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 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__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2)) , a__U21^#(X) -> c_10(X) , a__U21^#(tt()) -> c_11() , a__U31^#(X1, X2) -> c_12(X1, X2) , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2))) , a__U32^#(X) -> c_14(X) , a__U32^#(tt()) -> c_15() , a__U41^#(X1, X2) -> c_16(X1, X2) , a__U41^#(tt(), N) -> c_17(mark^#(N)) , mark^#(tt()) -> c_18() , mark^#(s(X)) -> c_19(mark^#(X)) , mark^#(0()) -> c_20() , mark^#(plus(X1, X2)) -> c_21(a__plus^#(mark(X1), mark(X2))) , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2))) , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2)) , mark^#(U12(X)) -> c_24(a__U12^#(mark(X))) , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) , mark^#(U21(X)) -> c_26(a__U21^#(mark(X))) , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2)) , mark^#(U32(X)) -> c_28(a__U32^#(mark(X))) , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2)) , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3)) , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3)) , mark^#(U61(X)) -> c_32(a__U61^#(mark(X))) , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3)) , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3)) , a__plus^#(X1, X2) -> c_39(X1, X2) , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N)) , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N)) , a__x^#(X1, X2) -> c_48(X1, X2) , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N)) , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N))) , a__U51^#(X1, X2, X3) -> c_35(X1, X2, X3) , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N)) , a__U52^#(X1, X2, X3) -> c_37(X1, X2, X3) , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M))) , a__U61^#(X) -> c_42(X) , a__U61^#(tt()) -> c_43() , a__U71^#(X1, X2, X3) -> c_44(X1, X2, X3) , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N)) , a__U72^#(X1, X2, X3) -> c_46(X1, X2, X3) , a__U72^#(tt(), M, N) -> c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 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__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) , a__U21(X) -> U21(X) , a__U21(tt()) -> tt() , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), V2) -> a__U32(a__isNat(V2)) , a__U32(X) -> U32(X) , a__U32(tt()) -> tt() , a__U41(X1, X2) -> U41(X1, X2) , a__U41(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(x(X1, X2)) -> a__x(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(U32(X)) -> a__U32(mark(X)) , mark(U41(X1, X2)) -> a__U41(mark(X1), X2) , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) , mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) , mark(U61(X)) -> a__U61(mark(X)) , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) , mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) , a__U51(X1, X2, X3) -> U51(X1, X2, X3) , a__U51(tt(), M, N) -> a__U52(a__isNat(N), M, N) , a__U52(X1, X2, X3) -> U52(X1, X2, X3) , a__U52(tt(), M, N) -> s(a__plus(mark(N), mark(M))) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) , a__plus(N, 0()) -> a__U41(a__isNat(N), N) , a__U61(X) -> U61(X) , a__U61(tt()) -> 0() , a__U71(X1, X2, X3) -> U71(X1, X2, X3) , a__U71(tt(), M, N) -> a__U72(a__isNat(N), M, N) , a__U72(X1, X2, X3) -> U72(X1, X2, X3) , a__U72(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__U71(a__isNat(M), M, N) , a__x(N, 0()) -> a__U61(a__isNat(N)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,7,11,15,18,20,46} by applications of Pre({4,7,11,15,18,20,46}) = {1,2,3,5,6,10,12,13,14,16,17,19,24,25,26,28,32,35,38,40,41,43,45,47,49}. 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__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2)) , 10: a__U21^#(X) -> c_10(X) , 11: a__U21^#(tt()) -> c_11() , 12: a__U31^#(X1, X2) -> c_12(X1, X2) , 13: a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2))) , 14: a__U32^#(X) -> c_14(X) , 15: a__U32^#(tt()) -> c_15() , 16: a__U41^#(X1, X2) -> c_16(X1, X2) , 17: a__U41^#(tt(), N) -> c_17(mark^#(N)) , 18: mark^#(tt()) -> c_18() , 19: mark^#(s(X)) -> c_19(mark^#(X)) , 20: mark^#(0()) -> c_20() , 21: mark^#(plus(X1, X2)) -> c_21(a__plus^#(mark(X1), mark(X2))) , 22: mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2))) , 23: mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2)) , 24: mark^#(U12(X)) -> c_24(a__U12^#(mark(X))) , 25: mark^#(isNat(X)) -> c_25(a__isNat^#(X)) , 26: mark^#(U21(X)) -> c_26(a__U21^#(mark(X))) , 27: mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2)) , 28: mark^#(U32(X)) -> c_28(a__U32^#(mark(X))) , 29: mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2)) , 30: mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3)) , 31: mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3)) , 32: mark^#(U61(X)) -> c_32(a__U61^#(mark(X))) , 33: mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3)) , 34: mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3)) , 35: a__plus^#(X1, X2) -> c_39(X1, X2) , 36: a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N)) , 37: a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N)) , 38: a__x^#(X1, X2) -> c_48(X1, X2) , 39: a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N)) , 40: a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N))) , 41: a__U51^#(X1, X2, X3) -> c_35(X1, X2, X3) , 42: a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N)) , 43: a__U52^#(X1, X2, X3) -> c_37(X1, X2, X3) , 44: a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M))) , 45: a__U61^#(X) -> c_42(X) , 46: a__U61^#(tt()) -> c_43() , 47: a__U71^#(X1, X2, X3) -> c_44(X1, X2, X3) , 48: a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N)) , 49: a__U72^#(X1, X2, X3) -> c_46(X1, X2, X3) , 50: a__U72^#(tt(), M, N) -> c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 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__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2)) , a__U21^#(X) -> c_10(X) , a__U31^#(X1, X2) -> c_12(X1, X2) , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2))) , a__U32^#(X) -> c_14(X) , a__U41^#(X1, X2) -> c_16(X1, X2) , a__U41^#(tt(), N) -> c_17(mark^#(N)) , mark^#(s(X)) -> c_19(mark^#(X)) , mark^#(plus(X1, X2)) -> c_21(a__plus^#(mark(X1), mark(X2))) , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2))) , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2)) , mark^#(U12(X)) -> c_24(a__U12^#(mark(X))) , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) , mark^#(U21(X)) -> c_26(a__U21^#(mark(X))) , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2)) , mark^#(U32(X)) -> c_28(a__U32^#(mark(X))) , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2)) , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3)) , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3)) , mark^#(U61(X)) -> c_32(a__U61^#(mark(X))) , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3)) , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3)) , a__plus^#(X1, X2) -> c_39(X1, X2) , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N)) , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N)) , a__x^#(X1, X2) -> c_48(X1, X2) , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N)) , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N))) , a__U51^#(X1, X2, X3) -> c_35(X1, X2, X3) , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N)) , a__U52^#(X1, X2, X3) -> c_37(X1, X2, X3) , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M))) , a__U61^#(X) -> c_42(X) , a__U71^#(X1, X2, X3) -> c_44(X1, X2, X3) , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N)) , a__U72^#(X1, X2, X3) -> c_46(X1, X2, X3) , a__U72^#(tt(), M, N) -> c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 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__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) , a__U21(X) -> U21(X) , a__U21(tt()) -> tt() , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), V2) -> a__U32(a__isNat(V2)) , a__U32(X) -> U32(X) , a__U32(tt()) -> tt() , a__U41(X1, X2) -> U41(X1, X2) , a__U41(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(x(X1, X2)) -> a__x(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(U32(X)) -> a__U32(mark(X)) , mark(U41(X1, X2)) -> a__U41(mark(X1), X2) , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) , mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) , mark(U61(X)) -> a__U61(mark(X)) , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) , mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) , a__U51(X1, X2, X3) -> U51(X1, X2, X3) , a__U51(tt(), M, N) -> a__U52(a__isNat(N), M, N) , a__U52(X1, X2, X3) -> U52(X1, X2, X3) , a__U52(tt(), M, N) -> s(a__plus(mark(N), mark(M))) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) , a__plus(N, 0()) -> a__U41(a__isNat(N), N) , a__U61(X) -> U61(X) , a__U61(tt()) -> 0() , a__U71(X1, X2, X3) -> U71(X1, X2, X3) , a__U71(tt(), M, N) -> a__U72(a__isNat(N), M, N) , a__U72(X1, X2, X3) -> U72(X1, X2, X3) , a__U72(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__U71(a__isNat(M), M, N) , a__x(N, 0()) -> a__U61(a__isNat(N)) } Weak DPs: { a__U12^#(tt()) -> c_4() , a__isNat^#(0()) -> c_7() , a__U21^#(tt()) -> c_11() , a__U32^#(tt()) -> c_15() , mark^#(tt()) -> c_18() , mark^#(0()) -> c_20() , a__U61^#(tt()) -> c_43() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..