MAYBE 1154.23/297.71 MAYBE 1154.23/297.71 1154.23/297.71 We are left with following problem, upon which TcT provides the 1154.23/297.71 certificate MAYBE. 1154.23/297.71 1154.23/297.71 Strict Trs: 1154.23/297.71 { a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1154.23/297.71 , a__U11(tt(), V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 1154.23/297.71 , a__U12(X1, X2, X3) -> U12(X1, X2, X3) 1154.23/297.71 , a__U12(tt(), V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 1154.23/297.71 , a__isNatKind(X) -> isNatKind(X) 1154.23/297.71 , a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 1154.23/297.71 , a__isNatKind(0()) -> tt() 1154.23/297.71 , a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 1154.23/297.71 , a__U13(X1, X2, X3) -> U13(X1, X2, X3) 1154.23/297.71 , a__U13(tt(), V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 1154.23/297.71 , a__U14(X1, X2, X3) -> U14(X1, X2, X3) 1154.23/297.71 , a__U14(tt(), V1, V2) -> a__U15(a__isNat(V1), V2) 1154.23/297.71 , a__U15(X1, X2) -> U15(X1, X2) 1154.23/297.71 , a__U15(tt(), V2) -> a__U16(a__isNat(V2)) 1154.23/297.71 , a__isNat(X) -> isNat(X) 1154.23/297.71 , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1154.23/297.71 , a__isNat(0()) -> tt() 1154.23/297.71 , a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 1154.23/297.71 , a__U16(X) -> U16(X) 1154.23/297.71 , a__U16(tt()) -> tt() 1154.23/297.71 , a__U21(X1, X2) -> U21(X1, X2) 1154.23/297.71 , a__U21(tt(), V1) -> a__U22(a__isNatKind(V1), V1) 1154.23/297.71 , a__U22(X1, X2) -> U22(X1, X2) 1154.23/297.71 , a__U22(tt(), V1) -> a__U23(a__isNat(V1)) 1154.23/297.71 , a__U23(X) -> U23(X) 1154.23/297.71 , a__U23(tt()) -> tt() 1154.23/297.71 , a__U31(X1, X2) -> U31(X1, X2) 1154.23/297.71 , a__U31(tt(), V2) -> a__U32(a__isNatKind(V2)) 1154.23/297.71 , a__U32(X) -> U32(X) 1154.23/297.71 , a__U32(tt()) -> tt() 1154.23/297.71 , a__U41(X) -> U41(X) 1154.23/297.71 , a__U41(tt()) -> tt() 1154.23/297.71 , a__U51(X1, X2) -> U51(X1, X2) 1154.23/297.71 , a__U51(tt(), N) -> a__U52(a__isNatKind(N), N) 1154.23/297.71 , a__U52(X1, X2) -> U52(X1, X2) 1154.23/297.71 , a__U52(tt(), N) -> mark(N) 1154.23/297.71 , mark(tt()) -> tt() 1154.23/297.71 , mark(s(X)) -> s(mark(X)) 1154.23/297.71 , mark(0()) -> 0() 1154.23/297.71 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1154.23/297.71 , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1154.23/297.71 , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 1154.23/297.71 , mark(isNatKind(X)) -> a__isNatKind(X) 1154.23/297.71 , mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 1154.23/297.71 , mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 1154.23/297.71 , mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 1154.23/297.71 , mark(isNat(X)) -> a__isNat(X) 1154.23/297.71 , mark(U16(X)) -> a__U16(mark(X)) 1154.23/297.71 , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1154.23/297.71 , mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1154.23/297.71 , mark(U23(X)) -> a__U23(mark(X)) 1154.23/297.71 , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1154.23/297.71 , mark(U32(X)) -> a__U32(mark(X)) 1154.23/297.71 , mark(U41(X)) -> a__U41(mark(X)) 1154.23/297.71 , mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1154.23/297.71 , mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 1154.23/297.71 , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 1154.23/297.71 , mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 1154.23/297.71 , mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 1154.23/297.71 , mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 1154.23/297.71 , a__U61(X1, X2, X3) -> U61(X1, X2, X3) 1154.23/297.71 , a__U61(tt(), M, N) -> a__U62(a__isNatKind(M), M, N) 1154.23/297.71 , a__U62(X1, X2, X3) -> U62(X1, X2, X3) 1154.23/297.71 , a__U62(tt(), M, N) -> a__U63(a__isNat(N), M, N) 1154.23/297.71 , a__U63(X1, X2, X3) -> U63(X1, X2, X3) 1154.23/297.71 , a__U63(tt(), M, N) -> a__U64(a__isNatKind(N), M, N) 1154.23/297.71 , a__U64(X1, X2, X3) -> U64(X1, X2, X3) 1154.23/297.71 , a__U64(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1154.23/297.71 , a__plus(X1, X2) -> plus(X1, X2) 1154.23/297.71 , a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 1154.23/297.71 , a__plus(N, 0()) -> a__U51(a__isNat(N), N) } 1154.23/297.71 Obligation: 1154.23/297.71 runtime complexity 1154.23/297.71 Answer: 1154.23/297.71 MAYBE 1154.23/297.71 1154.23/297.71 None of the processors succeeded. 1154.23/297.71 1154.23/297.71 Details of failed attempt(s): 1154.23/297.71 ----------------------------- 1154.23/297.71 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 1154.23/297.71 following reason: 1154.23/297.71 1154.23/297.71 Computation stopped due to timeout after 297.0 seconds. 1154.23/297.71 1154.23/297.71 2) 'Best' failed due to the following reason: 1154.23/297.71 1154.23/297.71 None of the processors succeeded. 1154.23/297.71 1154.23/297.71 Details of failed attempt(s): 1154.23/297.71 ----------------------------- 1154.23/297.71 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 1154.23/297.71 seconds)' failed due to the following reason: 1154.23/297.71 1154.23/297.71 Computation stopped due to timeout after 148.0 seconds. 1154.23/297.71 1154.23/297.71 2) 'Best' failed due to the following reason: 1154.23/297.71 1154.23/297.71 None of the processors succeeded. 1154.23/297.71 1154.23/297.71 Details of failed attempt(s): 1154.23/297.71 ----------------------------- 1154.23/297.71 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 1154.23/297.71 following reason: 1154.23/297.71 1154.23/297.71 The processor is inapplicable, reason: 1154.23/297.71 Processor only applicable for innermost runtime complexity analysis 1154.23/297.71 1154.23/297.71 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 1154.23/297.71 to the following reason: 1154.23/297.71 1154.23/297.71 The processor is inapplicable, reason: 1154.23/297.71 Processor only applicable for innermost runtime complexity analysis 1154.23/297.71 1154.23/297.71 1154.23/297.71 3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 1154.23/297.71 failed due to the following reason: 1154.23/297.71 1154.23/297.71 None of the processors succeeded. 1154.23/297.71 1154.23/297.71 Details of failed attempt(s): 1154.23/297.71 ----------------------------- 1154.23/297.71 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 1154.23/297.71 failed due to the following reason: 1154.23/297.71 1154.23/297.71 match-boundness of the problem could not be verified. 1154.23/297.71 1154.23/297.71 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 1154.23/297.71 failed due to the following reason: 1154.23/297.71 1154.23/297.71 match-boundness of the problem could not be verified. 1154.23/297.71 1154.23/297.71 1154.23/297.71 1154.23/297.71 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 1154.23/297.71 the following reason: 1154.23/297.71 1154.23/297.71 We add the following weak dependency pairs: 1154.23/297.71 1154.23/297.71 Strict DPs: 1154.23/297.71 { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1154.23/297.71 , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNatKind(V1), V1, V2)) 1154.23/297.71 , a__U12^#(X1, X2, X3) -> c_3(X1, X2, X3) 1154.23/297.71 , a__U12^#(tt(), V1, V2) -> c_4(a__U13^#(a__isNatKind(V2), V1, V2)) 1154.23/297.71 , a__U13^#(X1, X2, X3) -> c_9(X1, X2, X3) 1154.23/297.71 , a__U13^#(tt(), V1, V2) -> 1154.23/297.71 c_10(a__U14^#(a__isNatKind(V2), V1, V2)) 1154.23/297.71 , a__isNatKind^#(X) -> c_5(X) 1154.23/297.71 , a__isNatKind^#(s(V1)) -> c_6(a__U41^#(a__isNatKind(V1))) 1154.23/297.71 , a__isNatKind^#(0()) -> c_7() 1154.23/297.71 , a__isNatKind^#(plus(V1, V2)) -> 1154.23/297.71 c_8(a__U31^#(a__isNatKind(V1), V2)) 1154.23/297.71 , a__U41^#(X) -> c_31(X) 1154.23/297.71 , a__U41^#(tt()) -> c_32() 1154.23/297.71 , a__U31^#(X1, X2) -> c_27(X1, X2) 1154.23/297.71 , a__U31^#(tt(), V2) -> c_28(a__U32^#(a__isNatKind(V2))) 1154.23/297.71 , a__U14^#(X1, X2, X3) -> c_11(X1, X2, X3) 1154.23/297.71 , a__U14^#(tt(), V1, V2) -> c_12(a__U15^#(a__isNat(V1), V2)) 1154.23/297.71 , a__U15^#(X1, X2) -> c_13(X1, X2) 1154.23/297.71 , a__U15^#(tt(), V2) -> c_14(a__U16^#(a__isNat(V2))) 1154.23/297.71 , a__U16^#(X) -> c_19(X) 1154.23/297.71 , a__U16^#(tt()) -> c_20() 1154.23/297.71 , a__isNat^#(X) -> c_15(X) 1154.23/297.71 , a__isNat^#(s(V1)) -> c_16(a__U21^#(a__isNatKind(V1), V1)) 1154.23/297.71 , a__isNat^#(0()) -> c_17() 1154.23/297.71 , a__isNat^#(plus(V1, V2)) -> 1154.23/297.71 c_18(a__U11^#(a__isNatKind(V1), V1, V2)) 1154.23/297.71 , a__U21^#(X1, X2) -> c_21(X1, X2) 1154.23/297.71 , a__U21^#(tt(), V1) -> c_22(a__U22^#(a__isNatKind(V1), V1)) 1154.23/297.71 , a__U22^#(X1, X2) -> c_23(X1, X2) 1154.23/297.71 , a__U22^#(tt(), V1) -> c_24(a__U23^#(a__isNat(V1))) 1154.23/297.71 , a__U23^#(X) -> c_25(X) 1154.23/297.71 , a__U23^#(tt()) -> c_26() 1154.23/297.71 , a__U32^#(X) -> c_29(X) 1154.23/297.71 , a__U32^#(tt()) -> c_30() 1154.23/297.71 , a__U51^#(X1, X2) -> c_33(X1, X2) 1154.23/297.71 , a__U51^#(tt(), N) -> c_34(a__U52^#(a__isNatKind(N), N)) 1154.23/297.71 , a__U52^#(X1, X2) -> c_35(X1, X2) 1154.23/297.71 , a__U52^#(tt(), N) -> c_36(mark^#(N)) 1154.23/297.71 , mark^#(tt()) -> c_37() 1154.23/297.71 , mark^#(s(X)) -> c_38(mark^#(X)) 1154.23/297.71 , mark^#(0()) -> c_39() 1154.23/297.71 , mark^#(plus(X1, X2)) -> c_40(a__plus^#(mark(X1), mark(X2))) 1154.23/297.71 , mark^#(U11(X1, X2, X3)) -> c_41(a__U11^#(mark(X1), X2, X3)) 1154.23/297.71 , mark^#(U12(X1, X2, X3)) -> c_42(a__U12^#(mark(X1), X2, X3)) 1154.23/297.71 , mark^#(isNatKind(X)) -> c_43(a__isNatKind^#(X)) 1154.23/297.71 , mark^#(U13(X1, X2, X3)) -> c_44(a__U13^#(mark(X1), X2, X3)) 1154.23/297.71 , mark^#(U14(X1, X2, X3)) -> c_45(a__U14^#(mark(X1), X2, X3)) 1154.23/297.71 , mark^#(U15(X1, X2)) -> c_46(a__U15^#(mark(X1), X2)) 1154.23/297.71 , mark^#(isNat(X)) -> c_47(a__isNat^#(X)) 1154.23/297.71 , mark^#(U16(X)) -> c_48(a__U16^#(mark(X))) 1154.23/297.71 , mark^#(U21(X1, X2)) -> c_49(a__U21^#(mark(X1), X2)) 1154.23/297.71 , mark^#(U22(X1, X2)) -> c_50(a__U22^#(mark(X1), X2)) 1154.23/297.71 , mark^#(U23(X)) -> c_51(a__U23^#(mark(X))) 1154.23/297.71 , mark^#(U31(X1, X2)) -> c_52(a__U31^#(mark(X1), X2)) 1154.23/297.71 , mark^#(U32(X)) -> c_53(a__U32^#(mark(X))) 1154.23/297.71 , mark^#(U41(X)) -> c_54(a__U41^#(mark(X))) 1154.23/297.71 , mark^#(U51(X1, X2)) -> c_55(a__U51^#(mark(X1), X2)) 1154.23/297.71 , mark^#(U52(X1, X2)) -> c_56(a__U52^#(mark(X1), X2)) 1154.23/297.71 , mark^#(U61(X1, X2, X3)) -> c_57(a__U61^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U62(X1, X2, X3)) -> c_58(a__U62^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U63(X1, X2, X3)) -> c_59(a__U63^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U64(X1, X2, X3)) -> c_60(a__U64^#(mark(X1), X2, X3)) 1154.23/297.72 , a__plus^#(X1, X2) -> c_69(X1, X2) 1154.23/297.72 , a__plus^#(N, s(M)) -> c_70(a__U61^#(a__isNat(M), M, N)) 1154.23/297.72 , a__plus^#(N, 0()) -> c_71(a__U51^#(a__isNat(N), N)) 1154.23/297.72 , a__U61^#(X1, X2, X3) -> c_61(X1, X2, X3) 1154.23/297.72 , a__U61^#(tt(), M, N) -> c_62(a__U62^#(a__isNatKind(M), M, N)) 1154.23/297.72 , a__U62^#(X1, X2, X3) -> c_63(X1, X2, X3) 1154.23/297.72 , a__U62^#(tt(), M, N) -> c_64(a__U63^#(a__isNat(N), M, N)) 1154.23/297.72 , a__U63^#(X1, X2, X3) -> c_65(X1, X2, X3) 1154.23/297.72 , a__U63^#(tt(), M, N) -> c_66(a__U64^#(a__isNatKind(N), M, N)) 1154.23/297.72 , a__U64^#(X1, X2, X3) -> c_67(X1, X2, X3) 1154.23/297.72 , a__U64^#(tt(), M, N) -> c_68(a__plus^#(mark(N), mark(M))) } 1154.23/297.72 1154.23/297.72 and mark the set of starting terms. 1154.23/297.72 1154.23/297.72 We are left with following problem, upon which TcT provides the 1154.23/297.72 certificate MAYBE. 1154.23/297.72 1154.23/297.72 Strict DPs: 1154.23/297.72 { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1154.23/297.72 , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNatKind(V1), V1, V2)) 1154.23/297.72 , a__U12^#(X1, X2, X3) -> c_3(X1, X2, X3) 1154.23/297.72 , a__U12^#(tt(), V1, V2) -> c_4(a__U13^#(a__isNatKind(V2), V1, V2)) 1154.23/297.72 , a__U13^#(X1, X2, X3) -> c_9(X1, X2, X3) 1154.23/297.72 , a__U13^#(tt(), V1, V2) -> 1154.23/297.72 c_10(a__U14^#(a__isNatKind(V2), V1, V2)) 1154.23/297.72 , a__isNatKind^#(X) -> c_5(X) 1154.23/297.72 , a__isNatKind^#(s(V1)) -> c_6(a__U41^#(a__isNatKind(V1))) 1154.23/297.72 , a__isNatKind^#(0()) -> c_7() 1154.23/297.72 , a__isNatKind^#(plus(V1, V2)) -> 1154.23/297.72 c_8(a__U31^#(a__isNatKind(V1), V2)) 1154.23/297.72 , a__U41^#(X) -> c_31(X) 1154.23/297.72 , a__U41^#(tt()) -> c_32() 1154.23/297.72 , a__U31^#(X1, X2) -> c_27(X1, X2) 1154.23/297.72 , a__U31^#(tt(), V2) -> c_28(a__U32^#(a__isNatKind(V2))) 1154.23/297.72 , a__U14^#(X1, X2, X3) -> c_11(X1, X2, X3) 1154.23/297.72 , a__U14^#(tt(), V1, V2) -> c_12(a__U15^#(a__isNat(V1), V2)) 1154.23/297.72 , a__U15^#(X1, X2) -> c_13(X1, X2) 1154.23/297.72 , a__U15^#(tt(), V2) -> c_14(a__U16^#(a__isNat(V2))) 1154.23/297.72 , a__U16^#(X) -> c_19(X) 1154.23/297.72 , a__U16^#(tt()) -> c_20() 1154.23/297.72 , a__isNat^#(X) -> c_15(X) 1154.23/297.72 , a__isNat^#(s(V1)) -> c_16(a__U21^#(a__isNatKind(V1), V1)) 1154.23/297.72 , a__isNat^#(0()) -> c_17() 1154.23/297.72 , a__isNat^#(plus(V1, V2)) -> 1154.23/297.72 c_18(a__U11^#(a__isNatKind(V1), V1, V2)) 1154.23/297.72 , a__U21^#(X1, X2) -> c_21(X1, X2) 1154.23/297.72 , a__U21^#(tt(), V1) -> c_22(a__U22^#(a__isNatKind(V1), V1)) 1154.23/297.72 , a__U22^#(X1, X2) -> c_23(X1, X2) 1154.23/297.72 , a__U22^#(tt(), V1) -> c_24(a__U23^#(a__isNat(V1))) 1154.23/297.72 , a__U23^#(X) -> c_25(X) 1154.23/297.72 , a__U23^#(tt()) -> c_26() 1154.23/297.72 , a__U32^#(X) -> c_29(X) 1154.23/297.72 , a__U32^#(tt()) -> c_30() 1154.23/297.72 , a__U51^#(X1, X2) -> c_33(X1, X2) 1154.23/297.72 , a__U51^#(tt(), N) -> c_34(a__U52^#(a__isNatKind(N), N)) 1154.23/297.72 , a__U52^#(X1, X2) -> c_35(X1, X2) 1154.23/297.72 , a__U52^#(tt(), N) -> c_36(mark^#(N)) 1154.23/297.72 , mark^#(tt()) -> c_37() 1154.23/297.72 , mark^#(s(X)) -> c_38(mark^#(X)) 1154.23/297.72 , mark^#(0()) -> c_39() 1154.23/297.72 , mark^#(plus(X1, X2)) -> c_40(a__plus^#(mark(X1), mark(X2))) 1154.23/297.72 , mark^#(U11(X1, X2, X3)) -> c_41(a__U11^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U12(X1, X2, X3)) -> c_42(a__U12^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(isNatKind(X)) -> c_43(a__isNatKind^#(X)) 1154.23/297.72 , mark^#(U13(X1, X2, X3)) -> c_44(a__U13^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U14(X1, X2, X3)) -> c_45(a__U14^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U15(X1, X2)) -> c_46(a__U15^#(mark(X1), X2)) 1154.23/297.72 , mark^#(isNat(X)) -> c_47(a__isNat^#(X)) 1154.23/297.72 , mark^#(U16(X)) -> c_48(a__U16^#(mark(X))) 1154.23/297.72 , mark^#(U21(X1, X2)) -> c_49(a__U21^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U22(X1, X2)) -> c_50(a__U22^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U23(X)) -> c_51(a__U23^#(mark(X))) 1154.23/297.72 , mark^#(U31(X1, X2)) -> c_52(a__U31^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U32(X)) -> c_53(a__U32^#(mark(X))) 1154.23/297.72 , mark^#(U41(X)) -> c_54(a__U41^#(mark(X))) 1154.23/297.72 , mark^#(U51(X1, X2)) -> c_55(a__U51^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U52(X1, X2)) -> c_56(a__U52^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U61(X1, X2, X3)) -> c_57(a__U61^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U62(X1, X2, X3)) -> c_58(a__U62^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U63(X1, X2, X3)) -> c_59(a__U63^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U64(X1, X2, X3)) -> c_60(a__U64^#(mark(X1), X2, X3)) 1154.23/297.72 , a__plus^#(X1, X2) -> c_69(X1, X2) 1154.23/297.72 , a__plus^#(N, s(M)) -> c_70(a__U61^#(a__isNat(M), M, N)) 1154.23/297.72 , a__plus^#(N, 0()) -> c_71(a__U51^#(a__isNat(N), N)) 1154.23/297.72 , a__U61^#(X1, X2, X3) -> c_61(X1, X2, X3) 1154.23/297.72 , a__U61^#(tt(), M, N) -> c_62(a__U62^#(a__isNatKind(M), M, N)) 1154.23/297.72 , a__U62^#(X1, X2, X3) -> c_63(X1, X2, X3) 1154.23/297.72 , a__U62^#(tt(), M, N) -> c_64(a__U63^#(a__isNat(N), M, N)) 1154.23/297.72 , a__U63^#(X1, X2, X3) -> c_65(X1, X2, X3) 1154.23/297.72 , a__U63^#(tt(), M, N) -> c_66(a__U64^#(a__isNatKind(N), M, N)) 1154.23/297.72 , a__U64^#(X1, X2, X3) -> c_67(X1, X2, X3) 1154.23/297.72 , a__U64^#(tt(), M, N) -> c_68(a__plus^#(mark(N), mark(M))) } 1154.23/297.72 Strict Trs: 1154.23/297.72 { a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1154.23/297.72 , a__U11(tt(), V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 1154.23/297.72 , a__U12(X1, X2, X3) -> U12(X1, X2, X3) 1154.23/297.72 , a__U12(tt(), V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 1154.23/297.72 , a__isNatKind(X) -> isNatKind(X) 1154.23/297.72 , a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 1154.23/297.72 , a__isNatKind(0()) -> tt() 1154.23/297.72 , a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 1154.23/297.72 , a__U13(X1, X2, X3) -> U13(X1, X2, X3) 1154.23/297.72 , a__U13(tt(), V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 1154.23/297.72 , a__U14(X1, X2, X3) -> U14(X1, X2, X3) 1154.23/297.72 , a__U14(tt(), V1, V2) -> a__U15(a__isNat(V1), V2) 1154.23/297.72 , a__U15(X1, X2) -> U15(X1, X2) 1154.23/297.72 , a__U15(tt(), V2) -> a__U16(a__isNat(V2)) 1154.23/297.72 , a__isNat(X) -> isNat(X) 1154.23/297.72 , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1154.23/297.72 , a__isNat(0()) -> tt() 1154.23/297.72 , a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 1154.23/297.72 , a__U16(X) -> U16(X) 1154.23/297.72 , a__U16(tt()) -> tt() 1154.23/297.72 , a__U21(X1, X2) -> U21(X1, X2) 1154.23/297.72 , a__U21(tt(), V1) -> a__U22(a__isNatKind(V1), V1) 1154.23/297.72 , a__U22(X1, X2) -> U22(X1, X2) 1154.23/297.72 , a__U22(tt(), V1) -> a__U23(a__isNat(V1)) 1154.23/297.72 , a__U23(X) -> U23(X) 1154.23/297.72 , a__U23(tt()) -> tt() 1154.23/297.72 , a__U31(X1, X2) -> U31(X1, X2) 1154.23/297.72 , a__U31(tt(), V2) -> a__U32(a__isNatKind(V2)) 1154.23/297.72 , a__U32(X) -> U32(X) 1154.23/297.72 , a__U32(tt()) -> tt() 1154.23/297.72 , a__U41(X) -> U41(X) 1154.23/297.72 , a__U41(tt()) -> tt() 1154.23/297.72 , a__U51(X1, X2) -> U51(X1, X2) 1154.23/297.72 , a__U51(tt(), N) -> a__U52(a__isNatKind(N), N) 1154.23/297.72 , a__U52(X1, X2) -> U52(X1, X2) 1154.23/297.72 , a__U52(tt(), N) -> mark(N) 1154.23/297.72 , mark(tt()) -> tt() 1154.23/297.72 , mark(s(X)) -> s(mark(X)) 1154.23/297.72 , mark(0()) -> 0() 1154.23/297.72 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1154.23/297.72 , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1154.23/297.72 , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 1154.23/297.72 , mark(isNatKind(X)) -> a__isNatKind(X) 1154.23/297.72 , mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 1154.23/297.72 , mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 1154.23/297.72 , mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 1154.23/297.72 , mark(isNat(X)) -> a__isNat(X) 1154.23/297.72 , mark(U16(X)) -> a__U16(mark(X)) 1154.23/297.72 , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1154.23/297.72 , mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1154.23/297.72 , mark(U23(X)) -> a__U23(mark(X)) 1154.23/297.72 , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1154.23/297.72 , mark(U32(X)) -> a__U32(mark(X)) 1154.23/297.72 , mark(U41(X)) -> a__U41(mark(X)) 1154.23/297.72 , mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1154.23/297.72 , mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 1154.23/297.72 , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 1154.23/297.72 , mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 1154.23/297.72 , mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 1154.23/297.72 , mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 1154.23/297.72 , a__U61(X1, X2, X3) -> U61(X1, X2, X3) 1154.23/297.72 , a__U61(tt(), M, N) -> a__U62(a__isNatKind(M), M, N) 1154.23/297.72 , a__U62(X1, X2, X3) -> U62(X1, X2, X3) 1154.23/297.72 , a__U62(tt(), M, N) -> a__U63(a__isNat(N), M, N) 1154.23/297.72 , a__U63(X1, X2, X3) -> U63(X1, X2, X3) 1154.23/297.72 , a__U63(tt(), M, N) -> a__U64(a__isNatKind(N), M, N) 1154.23/297.72 , a__U64(X1, X2, X3) -> U64(X1, X2, X3) 1154.23/297.72 , a__U64(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1154.23/297.72 , a__plus(X1, X2) -> plus(X1, X2) 1154.23/297.72 , a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 1154.23/297.72 , a__plus(N, 0()) -> a__U51(a__isNat(N), N) } 1154.23/297.72 Obligation: 1154.23/297.72 runtime complexity 1154.23/297.72 Answer: 1154.23/297.72 MAYBE 1154.23/297.72 1154.23/297.72 We estimate the number of application of {9,12,20,23,30,32,37,39} 1154.23/297.72 by applications of Pre({9,12,20,23,30,32,37,39}) = 1154.23/297.72 {1,3,5,7,8,11,13,14,15,17,18,19,21,25,27,28,29,31,33,35,36,38,43,47,48,51,53,54,61,64,66,68,70}. 1154.23/297.72 Here rules are labeled as follows: 1154.23/297.72 1154.23/297.72 DPs: 1154.23/297.72 { 1: a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1154.23/297.72 , 2: a__U11^#(tt(), V1, V2) -> 1154.23/297.72 c_2(a__U12^#(a__isNatKind(V1), V1, V2)) 1154.23/297.72 , 3: a__U12^#(X1, X2, X3) -> c_3(X1, X2, X3) 1154.23/297.72 , 4: a__U12^#(tt(), V1, V2) -> 1154.23/297.72 c_4(a__U13^#(a__isNatKind(V2), V1, V2)) 1154.23/297.72 , 5: a__U13^#(X1, X2, X3) -> c_9(X1, X2, X3) 1154.23/297.72 , 6: a__U13^#(tt(), V1, V2) -> 1154.23/297.72 c_10(a__U14^#(a__isNatKind(V2), V1, V2)) 1154.23/297.72 , 7: a__isNatKind^#(X) -> c_5(X) 1154.23/297.72 , 8: a__isNatKind^#(s(V1)) -> c_6(a__U41^#(a__isNatKind(V1))) 1154.23/297.72 , 9: a__isNatKind^#(0()) -> c_7() 1154.23/297.72 , 10: a__isNatKind^#(plus(V1, V2)) -> 1154.23/297.72 c_8(a__U31^#(a__isNatKind(V1), V2)) 1154.23/297.72 , 11: a__U41^#(X) -> c_31(X) 1154.23/297.72 , 12: a__U41^#(tt()) -> c_32() 1154.23/297.72 , 13: a__U31^#(X1, X2) -> c_27(X1, X2) 1154.23/297.72 , 14: a__U31^#(tt(), V2) -> c_28(a__U32^#(a__isNatKind(V2))) 1154.23/297.72 , 15: a__U14^#(X1, X2, X3) -> c_11(X1, X2, X3) 1154.23/297.72 , 16: a__U14^#(tt(), V1, V2) -> c_12(a__U15^#(a__isNat(V1), V2)) 1154.23/297.72 , 17: a__U15^#(X1, X2) -> c_13(X1, X2) 1154.23/297.72 , 18: a__U15^#(tt(), V2) -> c_14(a__U16^#(a__isNat(V2))) 1154.23/297.72 , 19: a__U16^#(X) -> c_19(X) 1154.23/297.72 , 20: a__U16^#(tt()) -> c_20() 1154.23/297.72 , 21: a__isNat^#(X) -> c_15(X) 1154.23/297.72 , 22: a__isNat^#(s(V1)) -> c_16(a__U21^#(a__isNatKind(V1), V1)) 1154.23/297.72 , 23: a__isNat^#(0()) -> c_17() 1154.23/297.72 , 24: a__isNat^#(plus(V1, V2)) -> 1154.23/297.72 c_18(a__U11^#(a__isNatKind(V1), V1, V2)) 1154.23/297.72 , 25: a__U21^#(X1, X2) -> c_21(X1, X2) 1154.23/297.72 , 26: a__U21^#(tt(), V1) -> c_22(a__U22^#(a__isNatKind(V1), V1)) 1154.23/297.72 , 27: a__U22^#(X1, X2) -> c_23(X1, X2) 1154.23/297.72 , 28: a__U22^#(tt(), V1) -> c_24(a__U23^#(a__isNat(V1))) 1154.23/297.72 , 29: a__U23^#(X) -> c_25(X) 1154.23/297.72 , 30: a__U23^#(tt()) -> c_26() 1154.23/297.72 , 31: a__U32^#(X) -> c_29(X) 1154.23/297.72 , 32: a__U32^#(tt()) -> c_30() 1154.23/297.72 , 33: a__U51^#(X1, X2) -> c_33(X1, X2) 1154.23/297.72 , 34: a__U51^#(tt(), N) -> c_34(a__U52^#(a__isNatKind(N), N)) 1154.23/297.72 , 35: a__U52^#(X1, X2) -> c_35(X1, X2) 1154.23/297.72 , 36: a__U52^#(tt(), N) -> c_36(mark^#(N)) 1154.23/297.72 , 37: mark^#(tt()) -> c_37() 1154.23/297.72 , 38: mark^#(s(X)) -> c_38(mark^#(X)) 1154.23/297.72 , 39: mark^#(0()) -> c_39() 1154.23/297.72 , 40: mark^#(plus(X1, X2)) -> c_40(a__plus^#(mark(X1), mark(X2))) 1154.23/297.72 , 41: mark^#(U11(X1, X2, X3)) -> c_41(a__U11^#(mark(X1), X2, X3)) 1154.23/297.72 , 42: mark^#(U12(X1, X2, X3)) -> c_42(a__U12^#(mark(X1), X2, X3)) 1154.23/297.72 , 43: mark^#(isNatKind(X)) -> c_43(a__isNatKind^#(X)) 1154.23/297.72 , 44: mark^#(U13(X1, X2, X3)) -> c_44(a__U13^#(mark(X1), X2, X3)) 1154.23/297.72 , 45: mark^#(U14(X1, X2, X3)) -> c_45(a__U14^#(mark(X1), X2, X3)) 1154.23/297.72 , 46: mark^#(U15(X1, X2)) -> c_46(a__U15^#(mark(X1), X2)) 1154.23/297.72 , 47: mark^#(isNat(X)) -> c_47(a__isNat^#(X)) 1154.23/297.72 , 48: mark^#(U16(X)) -> c_48(a__U16^#(mark(X))) 1154.23/297.72 , 49: mark^#(U21(X1, X2)) -> c_49(a__U21^#(mark(X1), X2)) 1154.23/297.72 , 50: mark^#(U22(X1, X2)) -> c_50(a__U22^#(mark(X1), X2)) 1154.23/297.72 , 51: mark^#(U23(X)) -> c_51(a__U23^#(mark(X))) 1154.23/297.72 , 52: mark^#(U31(X1, X2)) -> c_52(a__U31^#(mark(X1), X2)) 1154.23/297.72 , 53: mark^#(U32(X)) -> c_53(a__U32^#(mark(X))) 1154.23/297.72 , 54: mark^#(U41(X)) -> c_54(a__U41^#(mark(X))) 1154.23/297.72 , 55: mark^#(U51(X1, X2)) -> c_55(a__U51^#(mark(X1), X2)) 1154.23/297.72 , 56: mark^#(U52(X1, X2)) -> c_56(a__U52^#(mark(X1), X2)) 1154.23/297.72 , 57: mark^#(U61(X1, X2, X3)) -> c_57(a__U61^#(mark(X1), X2, X3)) 1154.23/297.72 , 58: mark^#(U62(X1, X2, X3)) -> c_58(a__U62^#(mark(X1), X2, X3)) 1154.23/297.72 , 59: mark^#(U63(X1, X2, X3)) -> c_59(a__U63^#(mark(X1), X2, X3)) 1154.23/297.72 , 60: mark^#(U64(X1, X2, X3)) -> c_60(a__U64^#(mark(X1), X2, X3)) 1154.23/297.72 , 61: a__plus^#(X1, X2) -> c_69(X1, X2) 1154.23/297.72 , 62: a__plus^#(N, s(M)) -> c_70(a__U61^#(a__isNat(M), M, N)) 1154.23/297.72 , 63: a__plus^#(N, 0()) -> c_71(a__U51^#(a__isNat(N), N)) 1154.23/297.72 , 64: a__U61^#(X1, X2, X3) -> c_61(X1, X2, X3) 1154.23/297.72 , 65: a__U61^#(tt(), M, N) -> c_62(a__U62^#(a__isNatKind(M), M, N)) 1154.23/297.72 , 66: a__U62^#(X1, X2, X3) -> c_63(X1, X2, X3) 1154.23/297.72 , 67: a__U62^#(tt(), M, N) -> c_64(a__U63^#(a__isNat(N), M, N)) 1154.23/297.72 , 68: a__U63^#(X1, X2, X3) -> c_65(X1, X2, X3) 1154.23/297.72 , 69: a__U63^#(tt(), M, N) -> c_66(a__U64^#(a__isNatKind(N), M, N)) 1154.23/297.72 , 70: a__U64^#(X1, X2, X3) -> c_67(X1, X2, X3) 1154.23/297.72 , 71: a__U64^#(tt(), M, N) -> c_68(a__plus^#(mark(N), mark(M))) } 1154.23/297.72 1154.23/297.72 We are left with following problem, upon which TcT provides the 1154.23/297.72 certificate MAYBE. 1154.23/297.72 1154.23/297.72 Strict DPs: 1154.23/297.72 { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1154.23/297.72 , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNatKind(V1), V1, V2)) 1154.23/297.72 , a__U12^#(X1, X2, X3) -> c_3(X1, X2, X3) 1154.23/297.72 , a__U12^#(tt(), V1, V2) -> c_4(a__U13^#(a__isNatKind(V2), V1, V2)) 1154.23/297.72 , a__U13^#(X1, X2, X3) -> c_9(X1, X2, X3) 1154.23/297.72 , a__U13^#(tt(), V1, V2) -> 1154.23/297.72 c_10(a__U14^#(a__isNatKind(V2), V1, V2)) 1154.23/297.72 , a__isNatKind^#(X) -> c_5(X) 1154.23/297.72 , a__isNatKind^#(s(V1)) -> c_6(a__U41^#(a__isNatKind(V1))) 1154.23/297.72 , a__isNatKind^#(plus(V1, V2)) -> 1154.23/297.72 c_8(a__U31^#(a__isNatKind(V1), V2)) 1154.23/297.72 , a__U41^#(X) -> c_31(X) 1154.23/297.72 , a__U31^#(X1, X2) -> c_27(X1, X2) 1154.23/297.72 , a__U31^#(tt(), V2) -> c_28(a__U32^#(a__isNatKind(V2))) 1154.23/297.72 , a__U14^#(X1, X2, X3) -> c_11(X1, X2, X3) 1154.23/297.72 , a__U14^#(tt(), V1, V2) -> c_12(a__U15^#(a__isNat(V1), V2)) 1154.23/297.72 , a__U15^#(X1, X2) -> c_13(X1, X2) 1154.23/297.72 , a__U15^#(tt(), V2) -> c_14(a__U16^#(a__isNat(V2))) 1154.23/297.72 , a__U16^#(X) -> c_19(X) 1154.23/297.72 , a__isNat^#(X) -> c_15(X) 1154.23/297.72 , a__isNat^#(s(V1)) -> c_16(a__U21^#(a__isNatKind(V1), V1)) 1154.23/297.72 , a__isNat^#(plus(V1, V2)) -> 1154.23/297.72 c_18(a__U11^#(a__isNatKind(V1), V1, V2)) 1154.23/297.72 , a__U21^#(X1, X2) -> c_21(X1, X2) 1154.23/297.72 , a__U21^#(tt(), V1) -> c_22(a__U22^#(a__isNatKind(V1), V1)) 1154.23/297.72 , a__U22^#(X1, X2) -> c_23(X1, X2) 1154.23/297.72 , a__U22^#(tt(), V1) -> c_24(a__U23^#(a__isNat(V1))) 1154.23/297.72 , a__U23^#(X) -> c_25(X) 1154.23/297.72 , a__U32^#(X) -> c_29(X) 1154.23/297.72 , a__U51^#(X1, X2) -> c_33(X1, X2) 1154.23/297.72 , a__U51^#(tt(), N) -> c_34(a__U52^#(a__isNatKind(N), N)) 1154.23/297.72 , a__U52^#(X1, X2) -> c_35(X1, X2) 1154.23/297.72 , a__U52^#(tt(), N) -> c_36(mark^#(N)) 1154.23/297.72 , mark^#(s(X)) -> c_38(mark^#(X)) 1154.23/297.72 , mark^#(plus(X1, X2)) -> c_40(a__plus^#(mark(X1), mark(X2))) 1154.23/297.72 , mark^#(U11(X1, X2, X3)) -> c_41(a__U11^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U12(X1, X2, X3)) -> c_42(a__U12^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(isNatKind(X)) -> c_43(a__isNatKind^#(X)) 1154.23/297.72 , mark^#(U13(X1, X2, X3)) -> c_44(a__U13^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U14(X1, X2, X3)) -> c_45(a__U14^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U15(X1, X2)) -> c_46(a__U15^#(mark(X1), X2)) 1154.23/297.72 , mark^#(isNat(X)) -> c_47(a__isNat^#(X)) 1154.23/297.72 , mark^#(U16(X)) -> c_48(a__U16^#(mark(X))) 1154.23/297.72 , mark^#(U21(X1, X2)) -> c_49(a__U21^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U22(X1, X2)) -> c_50(a__U22^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U23(X)) -> c_51(a__U23^#(mark(X))) 1154.23/297.72 , mark^#(U31(X1, X2)) -> c_52(a__U31^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U32(X)) -> c_53(a__U32^#(mark(X))) 1154.23/297.72 , mark^#(U41(X)) -> c_54(a__U41^#(mark(X))) 1154.23/297.72 , mark^#(U51(X1, X2)) -> c_55(a__U51^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U52(X1, X2)) -> c_56(a__U52^#(mark(X1), X2)) 1154.23/297.72 , mark^#(U61(X1, X2, X3)) -> c_57(a__U61^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U62(X1, X2, X3)) -> c_58(a__U62^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U63(X1, X2, X3)) -> c_59(a__U63^#(mark(X1), X2, X3)) 1154.23/297.72 , mark^#(U64(X1, X2, X3)) -> c_60(a__U64^#(mark(X1), X2, X3)) 1154.23/297.72 , a__plus^#(X1, X2) -> c_69(X1, X2) 1154.23/297.72 , a__plus^#(N, s(M)) -> c_70(a__U61^#(a__isNat(M), M, N)) 1154.23/297.72 , a__plus^#(N, 0()) -> c_71(a__U51^#(a__isNat(N), N)) 1154.23/297.72 , a__U61^#(X1, X2, X3) -> c_61(X1, X2, X3) 1154.23/297.72 , a__U61^#(tt(), M, N) -> c_62(a__U62^#(a__isNatKind(M), M, N)) 1154.23/297.72 , a__U62^#(X1, X2, X3) -> c_63(X1, X2, X3) 1154.23/297.72 , a__U62^#(tt(), M, N) -> c_64(a__U63^#(a__isNat(N), M, N)) 1154.23/297.72 , a__U63^#(X1, X2, X3) -> c_65(X1, X2, X3) 1154.23/297.72 , a__U63^#(tt(), M, N) -> c_66(a__U64^#(a__isNatKind(N), M, N)) 1154.23/297.72 , a__U64^#(X1, X2, X3) -> c_67(X1, X2, X3) 1154.23/297.72 , a__U64^#(tt(), M, N) -> c_68(a__plus^#(mark(N), mark(M))) } 1154.23/297.72 Strict Trs: 1154.23/297.72 { a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1154.23/297.72 , a__U11(tt(), V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 1154.23/297.73 , a__U12(X1, X2, X3) -> U12(X1, X2, X3) 1154.23/297.73 , a__U12(tt(), V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 1154.23/297.73 , a__isNatKind(X) -> isNatKind(X) 1154.23/297.73 , a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 1154.23/297.73 , a__isNatKind(0()) -> tt() 1154.23/297.73 , a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 1154.23/297.73 , a__U13(X1, X2, X3) -> U13(X1, X2, X3) 1154.23/297.73 , a__U13(tt(), V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 1154.23/297.73 , a__U14(X1, X2, X3) -> U14(X1, X2, X3) 1154.23/297.73 , a__U14(tt(), V1, V2) -> a__U15(a__isNat(V1), V2) 1154.23/297.73 , a__U15(X1, X2) -> U15(X1, X2) 1154.23/297.73 , a__U15(tt(), V2) -> a__U16(a__isNat(V2)) 1154.23/297.73 , a__isNat(X) -> isNat(X) 1154.23/297.73 , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1154.23/297.73 , a__isNat(0()) -> tt() 1154.23/297.73 , a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 1154.23/297.73 , a__U16(X) -> U16(X) 1154.23/297.73 , a__U16(tt()) -> tt() 1154.23/297.73 , a__U21(X1, X2) -> U21(X1, X2) 1154.23/297.73 , a__U21(tt(), V1) -> a__U22(a__isNatKind(V1), V1) 1154.23/297.73 , a__U22(X1, X2) -> U22(X1, X2) 1154.23/297.73 , a__U22(tt(), V1) -> a__U23(a__isNat(V1)) 1154.23/297.73 , a__U23(X) -> U23(X) 1154.23/297.73 , a__U23(tt()) -> tt() 1154.23/297.73 , a__U31(X1, X2) -> U31(X1, X2) 1154.23/297.73 , a__U31(tt(), V2) -> a__U32(a__isNatKind(V2)) 1154.23/297.73 , a__U32(X) -> U32(X) 1154.23/297.73 , a__U32(tt()) -> tt() 1154.23/297.73 , a__U41(X) -> U41(X) 1154.23/297.73 , a__U41(tt()) -> tt() 1154.23/297.73 , a__U51(X1, X2) -> U51(X1, X2) 1154.23/297.73 , a__U51(tt(), N) -> a__U52(a__isNatKind(N), N) 1154.23/297.73 , a__U52(X1, X2) -> U52(X1, X2) 1154.23/297.73 , a__U52(tt(), N) -> mark(N) 1154.23/297.73 , mark(tt()) -> tt() 1154.23/297.73 , mark(s(X)) -> s(mark(X)) 1154.23/297.73 , mark(0()) -> 0() 1154.23/297.73 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1154.23/297.73 , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1154.23/297.73 , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 1154.23/297.73 , mark(isNatKind(X)) -> a__isNatKind(X) 1154.23/297.73 , mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 1154.23/297.73 , mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 1154.23/297.73 , mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 1154.23/297.73 , mark(isNat(X)) -> a__isNat(X) 1154.23/297.73 , mark(U16(X)) -> a__U16(mark(X)) 1154.23/297.73 , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1154.23/297.73 , mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1154.23/297.73 , mark(U23(X)) -> a__U23(mark(X)) 1154.23/297.73 , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1154.23/297.73 , mark(U32(X)) -> a__U32(mark(X)) 1154.23/297.73 , mark(U41(X)) -> a__U41(mark(X)) 1154.23/297.73 , mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1154.23/297.73 , mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 1154.23/297.73 , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 1154.23/297.73 , mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 1154.23/297.73 , mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 1154.23/297.73 , mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 1154.23/297.73 , a__U61(X1, X2, X3) -> U61(X1, X2, X3) 1154.23/297.73 , a__U61(tt(), M, N) -> a__U62(a__isNatKind(M), M, N) 1154.23/297.73 , a__U62(X1, X2, X3) -> U62(X1, X2, X3) 1154.23/297.73 , a__U62(tt(), M, N) -> a__U63(a__isNat(N), M, N) 1154.23/297.73 , a__U63(X1, X2, X3) -> U63(X1, X2, X3) 1154.23/297.73 , a__U63(tt(), M, N) -> a__U64(a__isNatKind(N), M, N) 1154.23/297.73 , a__U64(X1, X2, X3) -> U64(X1, X2, X3) 1154.23/297.73 , a__U64(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1154.23/297.73 , a__plus(X1, X2) -> plus(X1, X2) 1154.23/297.73 , a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 1154.23/297.73 , a__plus(N, 0()) -> a__U51(a__isNat(N), N) } 1154.23/297.73 Weak DPs: 1154.23/297.73 { a__isNatKind^#(0()) -> c_7() 1154.23/297.73 , a__U41^#(tt()) -> c_32() 1154.23/297.73 , a__U16^#(tt()) -> c_20() 1154.23/297.73 , a__isNat^#(0()) -> c_17() 1154.23/297.73 , a__U23^#(tt()) -> c_26() 1154.23/297.73 , a__U32^#(tt()) -> c_30() 1154.23/297.73 , mark^#(tt()) -> c_37() 1154.23/297.73 , mark^#(0()) -> c_39() } 1154.23/297.73 Obligation: 1154.23/297.73 runtime complexity 1154.23/297.73 Answer: 1154.23/297.73 MAYBE 1154.23/297.73 1154.23/297.73 Empty strict component of the problem is NOT empty. 1154.23/297.73 1154.23/297.73 1154.23/297.73 Arrrr.. 1155.09/298.40 EOF