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: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { a__U11^#(X1, X2) -> c_1() , a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2)), a__isNat^#(V2)) , a__U12^#(X) -> c_3() , a__U12^#(tt()) -> c_4() , a__isNat^#(X) -> c_5() , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1)), a__isNat^#(V1)) , a__isNat^#(0()) -> c_7() , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U21^#(X) -> c_10() , a__U21^#(tt()) -> c_11() , a__U31^#(X1, X2) -> c_12() , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2)), a__isNat^#(V2)) , a__U32^#(X) -> c_14() , a__U32^#(tt()) -> c_15() , a__U41^#(X1, X2) -> c_16() , 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^#(X1), mark^#(X2)) , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_24(a__U12^#(mark(X)), mark^#(X)) , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) , mark^#(U21(X)) -> c_26(a__U21^#(mark(X)), mark^#(X)) , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_28(a__U32^#(mark(X)), mark^#(X)) , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2), mark^#(X1)) , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U61(X)) -> c_32(a__U61^#(mark(X)), mark^#(X)) , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3), mark^#(X1)) , a__plus^#(X1, X2) -> c_39() , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N), a__isNat^#(M)) , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N), a__isNat^#(N)) , a__x^#(X1, X2) -> c_48() , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N), a__isNat^#(M)) , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N)), a__isNat^#(N)) , a__U51^#(X1, X2, X3) -> c_35() , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U52^#(X1, X2, X3) -> c_37() , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) , a__U61^#(X) -> c_42() , a__U61^#(tt()) -> c_43() , a__U71^#(X1, X2, X3) -> c_44() , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U72^#(X1, X2, X3) -> c_46() , a__U72^#(tt(), M, N) -> c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N)), a__x^#(mark(N), mark(M)), 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() , a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2)), a__isNat^#(V2)) , a__U12^#(X) -> c_3() , a__U12^#(tt()) -> c_4() , a__isNat^#(X) -> c_5() , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1)), a__isNat^#(V1)) , a__isNat^#(0()) -> c_7() , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U21^#(X) -> c_10() , a__U21^#(tt()) -> c_11() , a__U31^#(X1, X2) -> c_12() , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2)), a__isNat^#(V2)) , a__U32^#(X) -> c_14() , a__U32^#(tt()) -> c_15() , a__U41^#(X1, X2) -> c_16() , 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^#(X1), mark^#(X2)) , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_24(a__U12^#(mark(X)), mark^#(X)) , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) , mark^#(U21(X)) -> c_26(a__U21^#(mark(X)), mark^#(X)) , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_28(a__U32^#(mark(X)), mark^#(X)) , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2), mark^#(X1)) , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U61(X)) -> c_32(a__U61^#(mark(X)), mark^#(X)) , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3), mark^#(X1)) , a__plus^#(X1, X2) -> c_39() , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N), a__isNat^#(M)) , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N), a__isNat^#(N)) , a__x^#(X1, X2) -> c_48() , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N), a__isNat^#(M)) , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N)), a__isNat^#(N)) , a__U51^#(X1, X2, X3) -> c_35() , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U52^#(X1, X2, X3) -> c_37() , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) , a__U61^#(X) -> c_42() , a__U61^#(tt()) -> c_43() , a__U71^#(X1, X2, X3) -> c_44() , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U72^#(X1, X2, X3) -> c_46() , a__U72^#(tt(), M, N) -> c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N)), a__x^#(mark(N), mark(M)), mark^#(N), mark^#(M), mark^#(N)) } Weak 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: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,3,4,5,7,10,11,12,14,15,16,18,20,35,38,41,43,45,46,47,49} by applications of Pre({1,3,4,5,7,10,11,12,14,15,16,18,20,35,38,41,43,45,46,47,49}) = {2,6,8,9,13,17,19,21,22,23,24,25,26,27,28,29,30,31,32,33,34,36,37,39,40,42,44,48,50}. Here rules are labeled as follows: DPs: { 1: a__U11^#(X1, X2) -> c_1() , 2: a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2)), a__isNat^#(V2)) , 3: a__U12^#(X) -> c_3() , 4: a__U12^#(tt()) -> c_4() , 5: a__isNat^#(X) -> c_5() , 6: a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1)), a__isNat^#(V1)) , 7: a__isNat^#(0()) -> c_7() , 8: a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2), a__isNat^#(V1)) , 9: a__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2), a__isNat^#(V1)) , 10: a__U21^#(X) -> c_10() , 11: a__U21^#(tt()) -> c_11() , 12: a__U31^#(X1, X2) -> c_12() , 13: a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2)), a__isNat^#(V2)) , 14: a__U32^#(X) -> c_14() , 15: a__U32^#(tt()) -> c_15() , 16: a__U41^#(X1, X2) -> c_16() , 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)), mark^#(X1), mark^#(X2)) , 22: mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , 23: mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2), mark^#(X1)) , 24: mark^#(U12(X)) -> c_24(a__U12^#(mark(X)), mark^#(X)) , 25: mark^#(isNat(X)) -> c_25(a__isNat^#(X)) , 26: mark^#(U21(X)) -> c_26(a__U21^#(mark(X)), mark^#(X)) , 27: mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2), mark^#(X1)) , 28: mark^#(U32(X)) -> c_28(a__U32^#(mark(X)), mark^#(X)) , 29: mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2), mark^#(X1)) , 30: mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , 31: mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3), mark^#(X1)) , 32: mark^#(U61(X)) -> c_32(a__U61^#(mark(X)), mark^#(X)) , 33: mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3), mark^#(X1)) , 34: mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3), mark^#(X1)) , 35: a__plus^#(X1, X2) -> c_39() , 36: a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N), a__isNat^#(M)) , 37: a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N), a__isNat^#(N)) , 38: a__x^#(X1, X2) -> c_48() , 39: a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N), a__isNat^#(M)) , 40: a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N)), a__isNat^#(N)) , 41: a__U51^#(X1, X2, X3) -> c_35() , 42: a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N), a__isNat^#(N)) , 43: a__U52^#(X1, X2, X3) -> c_37() , 44: a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) , 45: a__U61^#(X) -> c_42() , 46: a__U61^#(tt()) -> c_43() , 47: a__U71^#(X1, X2, X3) -> c_44() , 48: a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N), a__isNat^#(N)) , 49: a__U72^#(X1, X2, X3) -> c_46() , 50: a__U72^#(tt(), M, N) -> c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N)), a__x^#(mark(N), mark(M)), mark^#(N), mark^#(M), mark^#(N)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2)), a__isNat^#(V2)) , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1)), a__isNat^#(V1)) , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2)), a__isNat^#(V2)) , 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^#(X1), mark^#(X2)) , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_24(a__U12^#(mark(X)), mark^#(X)) , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) , mark^#(U21(X)) -> c_26(a__U21^#(mark(X)), mark^#(X)) , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_28(a__U32^#(mark(X)), mark^#(X)) , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2), mark^#(X1)) , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U61(X)) -> c_32(a__U61^#(mark(X)), mark^#(X)) , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3), mark^#(X1)) , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N), a__isNat^#(M)) , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N), a__isNat^#(N)) , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N), a__isNat^#(M)) , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N)), a__isNat^#(N)) , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U72^#(tt(), M, N) -> c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N)), a__x^#(mark(N), mark(M)), mark^#(N), mark^#(M), mark^#(N)) } Weak DPs: { a__U11^#(X1, X2) -> c_1() , a__U12^#(X) -> c_3() , a__U12^#(tt()) -> c_4() , a__isNat^#(X) -> c_5() , a__isNat^#(0()) -> c_7() , a__U21^#(X) -> c_10() , a__U21^#(tt()) -> c_11() , a__U31^#(X1, X2) -> c_12() , a__U32^#(X) -> c_14() , a__U32^#(tt()) -> c_15() , a__U41^#(X1, X2) -> c_16() , mark^#(tt()) -> c_18() , mark^#(0()) -> c_20() , a__plus^#(X1, X2) -> c_39() , a__x^#(X1, X2) -> c_48() , a__U51^#(X1, X2, X3) -> c_35() , a__U52^#(X1, X2, X3) -> c_37() , a__U61^#(X) -> c_42() , a__U61^#(tt()) -> c_43() , a__U71^#(X1, X2, X3) -> c_44() , a__U72^#(X1, X2, X3) -> c_46() } Weak 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: innermost runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { a__U11^#(X1, X2) -> c_1() , a__U12^#(X) -> c_3() , a__U12^#(tt()) -> c_4() , a__isNat^#(X) -> c_5() , a__isNat^#(0()) -> c_7() , a__U21^#(X) -> c_10() , a__U21^#(tt()) -> c_11() , a__U31^#(X1, X2) -> c_12() , a__U32^#(X) -> c_14() , a__U32^#(tt()) -> c_15() , a__U41^#(X1, X2) -> c_16() , mark^#(tt()) -> c_18() , mark^#(0()) -> c_20() , a__plus^#(X1, X2) -> c_39() , a__x^#(X1, X2) -> c_48() , a__U51^#(X1, X2, X3) -> c_35() , a__U52^#(X1, X2, X3) -> c_37() , a__U61^#(X) -> c_42() , a__U61^#(tt()) -> c_43() , a__U71^#(X1, X2, X3) -> c_44() , a__U72^#(X1, X2, X3) -> c_46() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2)), a__isNat^#(V2)) , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1)), a__isNat^#(V1)) , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2)), a__isNat^#(V2)) , 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^#(X1), mark^#(X2)) , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_24(a__U12^#(mark(X)), mark^#(X)) , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) , mark^#(U21(X)) -> c_26(a__U21^#(mark(X)), mark^#(X)) , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_28(a__U32^#(mark(X)), mark^#(X)) , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2), mark^#(X1)) , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U61(X)) -> c_32(a__U61^#(mark(X)), mark^#(X)) , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3), mark^#(X1)) , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N), a__isNat^#(M)) , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N), a__isNat^#(N)) , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N), a__isNat^#(M)) , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N)), a__isNat^#(N)) , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U72^#(tt(), M, N) -> c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N)), a__x^#(mark(N), mark(M)), mark^#(N), mark^#(M), mark^#(N)) } Weak 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: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2)), a__isNat^#(V2)) , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1)), a__isNat^#(V1)) , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2)), a__isNat^#(V2)) , mark^#(U12(X)) -> c_24(a__U12^#(mark(X)), mark^#(X)) , mark^#(U21(X)) -> c_26(a__U21^#(mark(X)), mark^#(X)) , mark^#(U32(X)) -> c_28(a__U32^#(mark(X)), mark^#(X)) , mark^#(U61(X)) -> c_32(a__U61^#(mark(X)), mark^#(X)) , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N)), a__isNat^#(N)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(tt(), V2) -> c_1(a__isNat^#(V2)) , a__isNat^#(s(V1)) -> c_2(a__isNat^#(V1)) , a__isNat^#(plus(V1, V2)) -> c_3(a__U11^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__isNat^#(x(V1, V2)) -> c_4(a__U31^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U31^#(tt(), V2) -> c_5(a__isNat^#(V2)) , a__U41^#(tt(), N) -> c_6(mark^#(N)) , mark^#(s(X)) -> c_7(mark^#(X)) , mark^#(plus(X1, X2)) -> c_8(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(x(X1, X2)) -> c_9(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(U11(X1, X2)) -> c_10(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_11(mark^#(X)) , mark^#(isNat(X)) -> c_12(a__isNat^#(X)) , mark^#(U21(X)) -> c_13(mark^#(X)) , mark^#(U31(X1, X2)) -> c_14(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_15(mark^#(X)) , mark^#(U41(X1, X2)) -> c_16(a__U41^#(mark(X1), X2), mark^#(X1)) , mark^#(U51(X1, X2, X3)) -> c_17(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2, X3)) -> c_18(a__U52^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U61(X)) -> c_19(mark^#(X)) , mark^#(U71(X1, X2, X3)) -> c_20(a__U71^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U72(X1, X2, X3)) -> c_21(a__U72^#(mark(X1), X2, X3), mark^#(X1)) , a__plus^#(N, s(M)) -> c_22(a__U51^#(a__isNat(M), M, N), a__isNat^#(M)) , a__plus^#(N, 0()) -> c_23(a__U41^#(a__isNat(N), N), a__isNat^#(N)) , a__x^#(N, s(M)) -> c_24(a__U71^#(a__isNat(M), M, N), a__isNat^#(M)) , a__x^#(N, 0()) -> c_25(a__isNat^#(N)) , a__U51^#(tt(), M, N) -> c_26(a__U52^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U52^#(tt(), M, N) -> c_27(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) , a__U71^#(tt(), M, N) -> c_28(a__U72^#(a__isNat(N), M, N), a__isNat^#(N)) , a__U72^#(tt(), M, N) -> c_29(a__plus^#(a__x(mark(N), mark(M)), mark(N)), a__x^#(mark(N), mark(M)), mark^#(N), mark^#(M), mark^#(N)) } Weak 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: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..