MAYBE 1179.67/298.02 MAYBE 1179.67/298.02 1179.67/298.02 We are left with following problem, upon which TcT provides the 1179.67/298.02 certificate MAYBE. 1179.67/298.02 1179.67/298.02 Strict Trs: 1179.67/298.02 { a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1179.67/298.02 , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2) 1179.67/298.02 , a__U12(X1, X2) -> U12(X1, X2) 1179.67/298.02 , a__U12(tt(), V2) -> a__U13(a__isNat(V2)) 1179.67/298.02 , a__isNat(X) -> isNat(X) 1179.67/298.02 , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1179.67/298.02 , a__isNat(0()) -> tt() 1179.67/298.02 , a__isNat(plus(V1, V2)) -> 1179.67/298.02 a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 1179.67/298.02 , a__isNat(x(V1, V2)) -> 1179.67/298.02 a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 1179.67/298.02 , a__U13(X) -> U13(X) 1179.67/298.02 , a__U13(tt()) -> tt() 1179.67/298.02 , a__U21(X1, X2) -> U21(X1, X2) 1179.67/298.02 , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) 1179.67/298.02 , a__U22(X) -> U22(X) 1179.67/298.02 , a__U22(tt()) -> tt() 1179.67/298.02 , a__U31(X1, X2, X3) -> U31(X1, X2, X3) 1179.67/298.02 , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2) 1179.67/298.02 , a__U32(X1, X2) -> U32(X1, X2) 1179.67/298.02 , a__U32(tt(), V2) -> a__U33(a__isNat(V2)) 1179.67/298.02 , a__U33(X) -> U33(X) 1179.67/298.02 , a__U33(tt()) -> tt() 1179.67/298.02 , a__U41(X1, X2) -> U41(X1, X2) 1179.67/298.02 , a__U41(tt(), N) -> mark(N) 1179.67/298.02 , mark(tt()) -> tt() 1179.67/298.02 , mark(s(X)) -> s(mark(X)) 1179.67/298.02 , mark(0()) -> 0() 1179.67/298.02 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1179.67/298.02 , mark(isNatKind(X)) -> a__isNatKind(X) 1179.67/298.02 , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 1179.67/298.02 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 1179.67/298.02 , mark(isNat(X)) -> a__isNat(X) 1179.67/298.02 , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1179.67/298.02 , mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1179.67/298.02 , mark(U13(X)) -> a__U13(mark(X)) 1179.67/298.02 , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1179.67/298.02 , mark(U22(X)) -> a__U22(mark(X)) 1179.67/298.02 , mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 1179.67/298.02 , mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1179.67/298.02 , mark(U33(X)) -> a__U33(mark(X)) 1179.67/298.02 , mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1179.67/298.02 , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1179.67/298.02 , mark(U61(X)) -> a__U61(mark(X)) 1179.67/298.02 , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 1179.67/298.02 , a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1179.67/298.02 , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1179.67/298.02 , a__plus(X1, X2) -> plus(X1, X2) 1179.67/298.02 , a__plus(N, s(M)) -> 1179.67/298.02 a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), 1179.67/298.02 and(isNat(N), isNatKind(N))), 1179.67/298.02 M, 1179.67/298.02 N) 1179.67/298.02 , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) 1179.67/298.02 , a__U61(X) -> U61(X) 1179.67/298.02 , a__U61(tt()) -> 0() 1179.67/298.02 , a__U71(X1, X2, X3) -> U71(X1, X2, X3) 1179.67/298.02 , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 1179.67/298.02 , a__x(X1, X2) -> x(X1, X2) 1179.67/298.02 , a__x(N, s(M)) -> 1179.67/298.02 a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), 1179.67/298.02 and(isNat(N), isNatKind(N))), 1179.67/298.02 M, 1179.67/298.02 N) 1179.67/298.02 , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(N))) 1179.67/298.02 , a__and(X1, X2) -> and(X1, X2) 1179.67/298.02 , a__and(tt(), X) -> mark(X) 1179.67/298.02 , a__isNatKind(X) -> isNatKind(X) 1179.67/298.02 , a__isNatKind(s(V1)) -> a__isNatKind(V1) 1179.67/298.02 , a__isNatKind(0()) -> tt() 1179.67/298.02 , a__isNatKind(plus(V1, V2)) -> 1179.67/298.02 a__and(a__isNatKind(V1), isNatKind(V2)) 1179.67/298.02 , a__isNatKind(x(V1, V2)) -> 1179.67/298.02 a__and(a__isNatKind(V1), isNatKind(V2)) } 1179.67/298.02 Obligation: 1179.67/298.02 runtime complexity 1179.67/298.02 Answer: 1179.67/298.02 MAYBE 1179.67/298.02 1179.67/298.02 None of the processors succeeded. 1179.67/298.02 1179.67/298.02 Details of failed attempt(s): 1179.67/298.02 ----------------------------- 1179.67/298.02 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 1179.67/298.02 following reason: 1179.67/298.02 1179.67/298.02 Computation stopped due to timeout after 297.0 seconds. 1179.67/298.02 1179.67/298.02 2) 'Best' failed due to the following reason: 1179.67/298.02 1179.67/298.02 None of the processors succeeded. 1179.67/298.02 1179.67/298.02 Details of failed attempt(s): 1179.67/298.02 ----------------------------- 1179.67/298.02 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 1179.67/298.02 seconds)' failed due to the following reason: 1179.67/298.02 1179.67/298.02 Computation stopped due to timeout after 148.0 seconds. 1179.67/298.02 1179.67/298.02 2) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 1179.67/298.02 failed due to the following reason: 1179.67/298.02 1179.67/298.02 None of the processors succeeded. 1179.67/298.02 1179.67/298.02 Details of failed attempt(s): 1179.67/298.02 ----------------------------- 1179.67/298.02 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 1179.67/298.02 failed due to the following reason: 1179.67/298.02 1179.67/298.02 match-boundness of the problem could not be verified. 1179.67/298.02 1179.67/298.02 2) 'Bounds with minimal-enrichment and initial automaton 'match'' 1179.67/298.02 failed due to the following reason: 1179.67/298.02 1179.67/298.02 match-boundness of the problem could not be verified. 1179.67/298.02 1179.67/298.02 1179.67/298.02 3) 'Best' failed due to the following reason: 1179.67/298.02 1179.67/298.02 None of the processors succeeded. 1179.67/298.02 1179.67/298.02 Details of failed attempt(s): 1179.67/298.02 ----------------------------- 1179.67/298.02 1) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 1179.67/298.02 to the following reason: 1179.67/298.02 1179.67/298.02 The processor is inapplicable, reason: 1179.67/298.02 Processor only applicable for innermost runtime complexity analysis 1179.67/298.02 1179.67/298.02 2) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 1179.67/298.02 following reason: 1179.67/298.02 1179.67/298.02 The processor is inapplicable, reason: 1179.67/298.02 Processor only applicable for innermost runtime complexity analysis 1179.67/298.02 1179.67/298.02 1179.67/298.02 1179.67/298.02 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 1179.67/298.02 the following reason: 1179.67/298.02 1179.67/298.02 We add the following weak dependency pairs: 1179.67/298.02 1179.67/298.02 Strict DPs: 1179.67/298.02 { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1179.67/298.02 , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2)) 1179.67/298.02 , a__U12^#(X1, X2) -> c_3(X1, X2) 1179.67/298.02 , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2))) 1179.67/298.02 , a__U13^#(X) -> c_10(X) 1179.67/298.02 , a__U13^#(tt()) -> c_11() 1179.67/298.02 , a__isNat^#(X) -> c_5(X) 1179.67/298.02 , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1)) 1179.67/298.02 , a__isNat^#(0()) -> c_7() 1179.67/298.02 , a__isNat^#(plus(V1, V2)) -> 1179.67/298.02 c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1179.67/298.02 , a__isNat^#(x(V1, V2)) -> 1179.67/298.02 c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1179.67/298.02 , a__U21^#(X1, X2) -> c_12(X1, X2) 1179.67/298.02 , a__U21^#(tt(), V1) -> c_13(a__U22^#(a__isNat(V1))) 1179.67/298.02 , a__U31^#(X1, X2, X3) -> c_16(X1, X2, X3) 1179.67/298.02 , a__U31^#(tt(), V1, V2) -> c_17(a__U32^#(a__isNat(V1), V2)) 1179.67/298.02 , a__U22^#(X) -> c_14(X) 1179.67/298.02 , a__U22^#(tt()) -> c_15() 1179.67/298.02 , a__U32^#(X1, X2) -> c_18(X1, X2) 1179.67/298.02 , a__U32^#(tt(), V2) -> c_19(a__U33^#(a__isNat(V2))) 1179.67/298.02 , a__U33^#(X) -> c_20(X) 1179.67/298.02 , a__U33^#(tt()) -> c_21() 1179.67/298.02 , a__U41^#(X1, X2) -> c_22(X1, X2) 1179.67/298.02 , a__U41^#(tt(), N) -> c_23(mark^#(N)) 1179.67/298.02 , mark^#(tt()) -> c_24() 1179.67/298.02 , mark^#(s(X)) -> c_25(mark^#(X)) 1179.67/298.02 , mark^#(0()) -> c_26() 1179.67/298.02 , mark^#(plus(X1, X2)) -> c_27(a__plus^#(mark(X1), mark(X2))) 1179.67/298.02 , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X)) 1179.67/298.02 , mark^#(x(X1, X2)) -> c_29(a__x^#(mark(X1), mark(X2))) 1179.67/298.02 , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2)) 1179.67/298.02 , mark^#(isNat(X)) -> c_31(a__isNat^#(X)) 1179.67/298.02 , mark^#(U11(X1, X2, X3)) -> c_32(a__U11^#(mark(X1), X2, X3)) 1179.67/298.02 , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2)) 1179.67/298.02 , mark^#(U13(X)) -> c_34(a__U13^#(mark(X))) 1179.67/298.02 , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2)) 1179.67/298.02 , mark^#(U22(X)) -> c_36(a__U22^#(mark(X))) 1179.67/298.02 , mark^#(U31(X1, X2, X3)) -> c_37(a__U31^#(mark(X1), X2, X3)) 1179.67/298.02 , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2)) 1179.67/298.02 , mark^#(U33(X)) -> c_39(a__U33^#(mark(X))) 1179.67/298.02 , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2)) 1179.67/298.02 , mark^#(U51(X1, X2, X3)) -> c_41(a__U51^#(mark(X1), X2, X3)) 1179.67/298.02 , mark^#(U61(X)) -> c_42(a__U61^#(mark(X))) 1179.67/298.02 , mark^#(U71(X1, X2, X3)) -> c_43(a__U71^#(mark(X1), X2, X3)) 1179.67/298.02 , a__plus^#(X1, X2) -> c_46(X1, X2) 1179.67/298.02 , a__plus^#(N, s(M)) -> 1179.67/298.02 c_47(a__U51^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1179.67/298.02 and(isNat(N), isNatKind(N))), 1179.67/298.02 M, 1179.67/298.02 N)) 1179.67/298.02 , a__plus^#(N, 0()) -> 1179.67/298.02 c_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N)) 1179.67/298.02 , a__isNatKind^#(X) -> c_58(X) 1179.67/298.02 , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1)) 1179.67/298.02 , a__isNatKind^#(0()) -> c_60() 1179.67/298.02 , a__isNatKind^#(plus(V1, V2)) -> 1179.67/298.02 c_61(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1179.67/298.02 , a__isNatKind^#(x(V1, V2)) -> 1179.67/298.02 c_62(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1179.67/298.02 , a__x^#(X1, X2) -> c_53(X1, X2) 1179.67/298.02 , a__x^#(N, s(M)) -> 1179.67/298.02 c_54(a__U71^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1179.67/298.02 and(isNat(N), isNatKind(N))), 1179.67/298.02 M, 1179.67/298.03 N)) 1179.67/298.03 , a__x^#(N, 0()) -> 1179.67/298.03 c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N)))) 1179.67/298.03 , a__and^#(X1, X2) -> c_56(X1, X2) 1179.67/298.03 , a__and^#(tt(), X) -> c_57(mark^#(X)) 1179.67/298.03 , a__U51^#(X1, X2, X3) -> c_44(X1, X2, X3) 1179.67/298.03 , a__U51^#(tt(), M, N) -> c_45(a__plus^#(mark(N), mark(M))) 1179.67/298.03 , a__U61^#(X) -> c_49(X) 1179.67/298.03 , a__U61^#(tt()) -> c_50() 1179.67/298.03 , a__U71^#(X1, X2, X3) -> c_51(X1, X2, X3) 1179.67/298.03 , a__U71^#(tt(), M, N) -> 1179.67/298.03 c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 1179.67/298.03 1179.67/298.03 and mark the set of starting terms. 1179.67/298.03 1179.67/298.03 We are left with following problem, upon which TcT provides the 1179.67/298.03 certificate MAYBE. 1179.67/298.03 1179.67/298.03 Strict DPs: 1179.67/298.03 { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1179.67/298.03 , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2)) 1179.67/298.03 , a__U12^#(X1, X2) -> c_3(X1, X2) 1179.67/298.03 , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2))) 1179.67/298.03 , a__U13^#(X) -> c_10(X) 1179.67/298.03 , a__U13^#(tt()) -> c_11() 1179.67/298.03 , a__isNat^#(X) -> c_5(X) 1179.67/298.03 , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1)) 1179.67/298.03 , a__isNat^#(0()) -> c_7() 1179.67/298.03 , a__isNat^#(plus(V1, V2)) -> 1179.67/298.03 c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1179.67/298.03 , a__isNat^#(x(V1, V2)) -> 1179.67/298.03 c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1179.67/298.03 , a__U21^#(X1, X2) -> c_12(X1, X2) 1179.67/298.03 , a__U21^#(tt(), V1) -> c_13(a__U22^#(a__isNat(V1))) 1179.67/298.03 , a__U31^#(X1, X2, X3) -> c_16(X1, X2, X3) 1179.67/298.03 , a__U31^#(tt(), V1, V2) -> c_17(a__U32^#(a__isNat(V1), V2)) 1179.67/298.03 , a__U22^#(X) -> c_14(X) 1179.67/298.03 , a__U22^#(tt()) -> c_15() 1179.67/298.03 , a__U32^#(X1, X2) -> c_18(X1, X2) 1179.67/298.03 , a__U32^#(tt(), V2) -> c_19(a__U33^#(a__isNat(V2))) 1179.67/298.03 , a__U33^#(X) -> c_20(X) 1179.67/298.03 , a__U33^#(tt()) -> c_21() 1179.67/298.03 , a__U41^#(X1, X2) -> c_22(X1, X2) 1179.67/298.03 , a__U41^#(tt(), N) -> c_23(mark^#(N)) 1179.67/298.03 , mark^#(tt()) -> c_24() 1179.67/298.03 , mark^#(s(X)) -> c_25(mark^#(X)) 1179.67/298.03 , mark^#(0()) -> c_26() 1179.67/298.03 , mark^#(plus(X1, X2)) -> c_27(a__plus^#(mark(X1), mark(X2))) 1179.67/298.03 , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X)) 1179.67/298.03 , mark^#(x(X1, X2)) -> c_29(a__x^#(mark(X1), mark(X2))) 1179.67/298.03 , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2)) 1179.67/298.03 , mark^#(isNat(X)) -> c_31(a__isNat^#(X)) 1179.67/298.03 , mark^#(U11(X1, X2, X3)) -> c_32(a__U11^#(mark(X1), X2, X3)) 1179.67/298.03 , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2)) 1179.67/298.03 , mark^#(U13(X)) -> c_34(a__U13^#(mark(X))) 1179.67/298.03 , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2)) 1179.67/298.03 , mark^#(U22(X)) -> c_36(a__U22^#(mark(X))) 1179.67/298.03 , mark^#(U31(X1, X2, X3)) -> c_37(a__U31^#(mark(X1), X2, X3)) 1179.67/298.03 , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2)) 1179.67/298.03 , mark^#(U33(X)) -> c_39(a__U33^#(mark(X))) 1179.67/298.03 , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2)) 1179.67/298.03 , mark^#(U51(X1, X2, X3)) -> c_41(a__U51^#(mark(X1), X2, X3)) 1179.67/298.03 , mark^#(U61(X)) -> c_42(a__U61^#(mark(X))) 1179.67/298.03 , mark^#(U71(X1, X2, X3)) -> c_43(a__U71^#(mark(X1), X2, X3)) 1179.67/298.03 , a__plus^#(X1, X2) -> c_46(X1, X2) 1179.67/298.03 , a__plus^#(N, s(M)) -> 1179.67/298.03 c_47(a__U51^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1179.67/298.03 and(isNat(N), isNatKind(N))), 1179.67/298.03 M, 1179.67/298.03 N)) 1179.67/298.03 , a__plus^#(N, 0()) -> 1179.67/298.03 c_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N)) 1179.67/298.03 , a__isNatKind^#(X) -> c_58(X) 1179.67/298.03 , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1)) 1179.67/298.03 , a__isNatKind^#(0()) -> c_60() 1179.67/298.03 , a__isNatKind^#(plus(V1, V2)) -> 1179.67/298.03 c_61(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1179.67/298.03 , a__isNatKind^#(x(V1, V2)) -> 1179.67/298.03 c_62(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1179.67/298.03 , a__x^#(X1, X2) -> c_53(X1, X2) 1179.67/298.03 , a__x^#(N, s(M)) -> 1179.67/298.03 c_54(a__U71^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1179.67/298.03 and(isNat(N), isNatKind(N))), 1179.67/298.03 M, 1179.67/298.03 N)) 1179.67/298.03 , a__x^#(N, 0()) -> 1179.67/298.03 c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N)))) 1179.67/298.03 , a__and^#(X1, X2) -> c_56(X1, X2) 1179.67/298.03 , a__and^#(tt(), X) -> c_57(mark^#(X)) 1179.67/298.03 , a__U51^#(X1, X2, X3) -> c_44(X1, X2, X3) 1179.67/298.03 , a__U51^#(tt(), M, N) -> c_45(a__plus^#(mark(N), mark(M))) 1179.67/298.03 , a__U61^#(X) -> c_49(X) 1179.67/298.03 , a__U61^#(tt()) -> c_50() 1179.67/298.03 , a__U71^#(X1, X2, X3) -> c_51(X1, X2, X3) 1179.67/298.03 , a__U71^#(tt(), M, N) -> 1179.67/298.03 c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 1179.67/298.03 Strict Trs: 1179.67/298.03 { a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1179.67/298.03 , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2) 1179.67/298.03 , a__U12(X1, X2) -> U12(X1, X2) 1179.67/298.03 , a__U12(tt(), V2) -> a__U13(a__isNat(V2)) 1179.67/298.03 , a__isNat(X) -> isNat(X) 1179.67/298.03 , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1179.67/298.03 , a__isNat(0()) -> tt() 1179.67/298.03 , a__isNat(plus(V1, V2)) -> 1179.67/298.03 a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 1179.67/298.03 , a__isNat(x(V1, V2)) -> 1179.67/298.03 a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 1179.67/298.03 , a__U13(X) -> U13(X) 1179.67/298.03 , a__U13(tt()) -> tt() 1179.67/298.03 , a__U21(X1, X2) -> U21(X1, X2) 1179.67/298.03 , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) 1179.67/298.03 , a__U22(X) -> U22(X) 1179.67/298.03 , a__U22(tt()) -> tt() 1179.67/298.03 , a__U31(X1, X2, X3) -> U31(X1, X2, X3) 1179.67/298.03 , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2) 1179.67/298.03 , a__U32(X1, X2) -> U32(X1, X2) 1179.67/298.03 , a__U32(tt(), V2) -> a__U33(a__isNat(V2)) 1179.67/298.03 , a__U33(X) -> U33(X) 1179.67/298.03 , a__U33(tt()) -> tt() 1179.67/298.03 , a__U41(X1, X2) -> U41(X1, X2) 1179.67/298.03 , a__U41(tt(), N) -> mark(N) 1179.67/298.03 , mark(tt()) -> tt() 1179.67/298.03 , mark(s(X)) -> s(mark(X)) 1179.67/298.03 , mark(0()) -> 0() 1179.67/298.03 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1179.67/298.03 , mark(isNatKind(X)) -> a__isNatKind(X) 1179.67/298.03 , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 1179.67/298.03 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 1179.67/298.03 , mark(isNat(X)) -> a__isNat(X) 1179.67/298.03 , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1179.67/298.03 , mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1179.67/298.03 , mark(U13(X)) -> a__U13(mark(X)) 1179.67/298.03 , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1179.67/298.03 , mark(U22(X)) -> a__U22(mark(X)) 1179.67/298.03 , mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 1179.67/298.03 , mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1179.67/298.03 , mark(U33(X)) -> a__U33(mark(X)) 1179.67/298.03 , mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1179.67/298.03 , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1179.67/298.03 , mark(U61(X)) -> a__U61(mark(X)) 1179.67/298.03 , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 1179.67/298.03 , a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1179.67/298.03 , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1179.67/298.03 , a__plus(X1, X2) -> plus(X1, X2) 1179.67/298.03 , a__plus(N, s(M)) -> 1179.67/298.03 a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), 1179.67/298.03 and(isNat(N), isNatKind(N))), 1179.67/298.03 M, 1179.67/298.03 N) 1179.67/298.03 , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) 1179.67/298.03 , a__U61(X) -> U61(X) 1179.67/298.03 , a__U61(tt()) -> 0() 1179.67/298.03 , a__U71(X1, X2, X3) -> U71(X1, X2, X3) 1179.67/298.03 , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 1179.67/298.03 , a__x(X1, X2) -> x(X1, X2) 1179.67/298.03 , a__x(N, s(M)) -> 1179.67/298.03 a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), 1179.67/298.03 and(isNat(N), isNatKind(N))), 1179.67/298.03 M, 1179.67/298.03 N) 1179.67/298.03 , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(N))) 1179.67/298.03 , a__and(X1, X2) -> and(X1, X2) 1179.67/298.03 , a__and(tt(), X) -> mark(X) 1179.67/298.03 , a__isNatKind(X) -> isNatKind(X) 1179.67/298.03 , a__isNatKind(s(V1)) -> a__isNatKind(V1) 1179.67/298.03 , a__isNatKind(0()) -> tt() 1179.67/298.03 , a__isNatKind(plus(V1, V2)) -> 1179.67/298.03 a__and(a__isNatKind(V1), isNatKind(V2)) 1179.67/298.03 , a__isNatKind(x(V1, V2)) -> 1179.67/298.03 a__and(a__isNatKind(V1), isNatKind(V2)) } 1179.67/298.03 Obligation: 1179.67/298.03 runtime complexity 1179.67/298.03 Answer: 1179.67/298.03 MAYBE 1179.67/298.03 1179.67/298.03 We estimate the number of application of {6,9,17,21,24,26,49,60} by 1179.67/298.03 applications of Pre({6,9,17,21,24,26,49,60}) = 1179.67/298.03 {1,3,4,5,7,12,13,14,16,18,19,20,22,23,25,28,31,34,36,39,42,44,47,48,52,54,55,56,57,59,61}. 1179.67/298.03 Here rules are labeled as follows: 1179.67/298.03 1179.67/298.03 DPs: 1179.67/298.03 { 1: a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1179.67/298.03 , 2: a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2)) 1179.67/298.03 , 3: a__U12^#(X1, X2) -> c_3(X1, X2) 1179.67/298.03 , 4: a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2))) 1179.67/298.03 , 5: a__U13^#(X) -> c_10(X) 1179.67/298.03 , 6: a__U13^#(tt()) -> c_11() 1179.67/298.03 , 7: a__isNat^#(X) -> c_5(X) 1179.67/298.03 , 8: a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1)) 1179.67/298.03 , 9: a__isNat^#(0()) -> c_7() 1180.43/298.22 , 10: a__isNat^#(plus(V1, V2)) -> 1180.43/298.22 c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1180.43/298.22 , 11: a__isNat^#(x(V1, V2)) -> 1180.43/298.22 c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1180.43/298.22 , 12: a__U21^#(X1, X2) -> c_12(X1, X2) 1180.43/298.22 , 13: a__U21^#(tt(), V1) -> c_13(a__U22^#(a__isNat(V1))) 1180.43/298.22 , 14: a__U31^#(X1, X2, X3) -> c_16(X1, X2, X3) 1180.43/298.22 , 15: a__U31^#(tt(), V1, V2) -> c_17(a__U32^#(a__isNat(V1), V2)) 1180.43/298.22 , 16: a__U22^#(X) -> c_14(X) 1180.43/298.22 , 17: a__U22^#(tt()) -> c_15() 1180.43/298.22 , 18: a__U32^#(X1, X2) -> c_18(X1, X2) 1180.43/298.22 , 19: a__U32^#(tt(), V2) -> c_19(a__U33^#(a__isNat(V2))) 1180.43/298.22 , 20: a__U33^#(X) -> c_20(X) 1180.43/298.22 , 21: a__U33^#(tt()) -> c_21() 1180.43/298.22 , 22: a__U41^#(X1, X2) -> c_22(X1, X2) 1180.43/298.22 , 23: a__U41^#(tt(), N) -> c_23(mark^#(N)) 1180.43/298.22 , 24: mark^#(tt()) -> c_24() 1180.43/298.22 , 25: mark^#(s(X)) -> c_25(mark^#(X)) 1180.43/298.22 , 26: mark^#(0()) -> c_26() 1180.43/298.22 , 27: mark^#(plus(X1, X2)) -> c_27(a__plus^#(mark(X1), mark(X2))) 1180.43/298.22 , 28: mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X)) 1180.43/298.22 , 29: mark^#(x(X1, X2)) -> c_29(a__x^#(mark(X1), mark(X2))) 1180.43/298.22 , 30: mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2)) 1180.43/298.22 , 31: mark^#(isNat(X)) -> c_31(a__isNat^#(X)) 1180.43/298.22 , 32: mark^#(U11(X1, X2, X3)) -> c_32(a__U11^#(mark(X1), X2, X3)) 1180.43/298.22 , 33: mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2)) 1180.43/298.22 , 34: mark^#(U13(X)) -> c_34(a__U13^#(mark(X))) 1180.43/298.22 , 35: mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2)) 1180.43/298.22 , 36: mark^#(U22(X)) -> c_36(a__U22^#(mark(X))) 1180.43/298.22 , 37: mark^#(U31(X1, X2, X3)) -> c_37(a__U31^#(mark(X1), X2, X3)) 1180.43/298.22 , 38: mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2)) 1180.43/298.22 , 39: mark^#(U33(X)) -> c_39(a__U33^#(mark(X))) 1180.43/298.22 , 40: mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2)) 1180.43/298.22 , 41: mark^#(U51(X1, X2, X3)) -> c_41(a__U51^#(mark(X1), X2, X3)) 1180.43/298.22 , 42: mark^#(U61(X)) -> c_42(a__U61^#(mark(X))) 1180.43/298.22 , 43: mark^#(U71(X1, X2, X3)) -> c_43(a__U71^#(mark(X1), X2, X3)) 1180.43/298.22 , 44: a__plus^#(X1, X2) -> c_46(X1, X2) 1180.43/298.22 , 45: a__plus^#(N, s(M)) -> 1180.43/298.22 c_47(a__U51^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1180.43/298.22 and(isNat(N), isNatKind(N))), 1180.43/298.22 M, 1180.43/298.22 N)) 1180.43/298.22 , 46: a__plus^#(N, 0()) -> 1180.43/298.22 c_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N)) 1180.43/298.22 , 47: a__isNatKind^#(X) -> c_58(X) 1180.43/298.22 , 48: a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1)) 1180.43/298.22 , 49: a__isNatKind^#(0()) -> c_60() 1180.43/298.22 , 50: a__isNatKind^#(plus(V1, V2)) -> 1180.43/298.22 c_61(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1180.43/298.22 , 51: a__isNatKind^#(x(V1, V2)) -> 1180.43/298.22 c_62(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1180.43/298.22 , 52: a__x^#(X1, X2) -> c_53(X1, X2) 1180.43/298.22 , 53: a__x^#(N, s(M)) -> 1180.43/298.22 c_54(a__U71^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1180.43/298.22 and(isNat(N), isNatKind(N))), 1180.43/298.22 M, 1180.43/298.22 N)) 1180.43/298.22 , 54: a__x^#(N, 0()) -> 1180.43/298.22 c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N)))) 1180.43/298.22 , 55: a__and^#(X1, X2) -> c_56(X1, X2) 1180.43/298.22 , 56: a__and^#(tt(), X) -> c_57(mark^#(X)) 1180.43/298.22 , 57: a__U51^#(X1, X2, X3) -> c_44(X1, X2, X3) 1180.43/298.22 , 58: a__U51^#(tt(), M, N) -> c_45(a__plus^#(mark(N), mark(M))) 1180.43/298.22 , 59: a__U61^#(X) -> c_49(X) 1180.43/298.22 , 60: a__U61^#(tt()) -> c_50() 1180.43/298.22 , 61: a__U71^#(X1, X2, X3) -> c_51(X1, X2, X3) 1180.43/298.22 , 62: a__U71^#(tt(), M, N) -> 1180.43/298.22 c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 1180.43/298.22 1180.43/298.22 We are left with following problem, upon which TcT provides the 1180.43/298.22 certificate MAYBE. 1180.43/298.22 1180.43/298.22 Strict DPs: 1180.43/298.22 { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1180.43/298.22 , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2)) 1180.43/298.22 , a__U12^#(X1, X2) -> c_3(X1, X2) 1180.43/298.22 , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2))) 1180.43/298.22 , a__U13^#(X) -> c_10(X) 1180.43/298.22 , a__isNat^#(X) -> c_5(X) 1180.43/298.22 , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1)) 1180.43/298.22 , a__isNat^#(plus(V1, V2)) -> 1180.43/298.22 c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1180.43/298.22 , a__isNat^#(x(V1, V2)) -> 1180.43/298.22 c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1180.43/298.22 , a__U21^#(X1, X2) -> c_12(X1, X2) 1180.43/298.22 , a__U21^#(tt(), V1) -> c_13(a__U22^#(a__isNat(V1))) 1180.43/298.22 , a__U31^#(X1, X2, X3) -> c_16(X1, X2, X3) 1180.43/298.22 , a__U31^#(tt(), V1, V2) -> c_17(a__U32^#(a__isNat(V1), V2)) 1180.43/298.22 , a__U22^#(X) -> c_14(X) 1180.43/298.22 , a__U32^#(X1, X2) -> c_18(X1, X2) 1180.43/298.22 , a__U32^#(tt(), V2) -> c_19(a__U33^#(a__isNat(V2))) 1180.43/298.22 , a__U33^#(X) -> c_20(X) 1180.43/298.22 , a__U41^#(X1, X2) -> c_22(X1, X2) 1180.43/298.22 , a__U41^#(tt(), N) -> c_23(mark^#(N)) 1180.43/298.22 , mark^#(s(X)) -> c_25(mark^#(X)) 1180.43/298.22 , mark^#(plus(X1, X2)) -> c_27(a__plus^#(mark(X1), mark(X2))) 1180.43/298.22 , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X)) 1180.43/298.22 , mark^#(x(X1, X2)) -> c_29(a__x^#(mark(X1), mark(X2))) 1180.43/298.22 , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2)) 1180.43/298.22 , mark^#(isNat(X)) -> c_31(a__isNat^#(X)) 1180.43/298.22 , mark^#(U11(X1, X2, X3)) -> c_32(a__U11^#(mark(X1), X2, X3)) 1180.43/298.22 , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2)) 1180.43/298.22 , mark^#(U13(X)) -> c_34(a__U13^#(mark(X))) 1180.43/298.22 , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2)) 1180.43/298.22 , mark^#(U22(X)) -> c_36(a__U22^#(mark(X))) 1180.43/298.22 , mark^#(U31(X1, X2, X3)) -> c_37(a__U31^#(mark(X1), X2, X3)) 1180.43/298.22 , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2)) 1180.43/298.22 , mark^#(U33(X)) -> c_39(a__U33^#(mark(X))) 1180.43/298.22 , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2)) 1180.43/298.22 , mark^#(U51(X1, X2, X3)) -> c_41(a__U51^#(mark(X1), X2, X3)) 1180.43/298.22 , mark^#(U61(X)) -> c_42(a__U61^#(mark(X))) 1180.43/298.22 , mark^#(U71(X1, X2, X3)) -> c_43(a__U71^#(mark(X1), X2, X3)) 1180.43/298.22 , a__plus^#(X1, X2) -> c_46(X1, X2) 1180.43/298.22 , a__plus^#(N, s(M)) -> 1180.43/298.22 c_47(a__U51^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1180.43/298.22 and(isNat(N), isNatKind(N))), 1180.43/298.22 M, 1180.43/298.22 N)) 1180.43/298.22 , a__plus^#(N, 0()) -> 1180.43/298.22 c_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N)) 1180.43/298.22 , a__isNatKind^#(X) -> c_58(X) 1180.43/298.22 , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1)) 1180.43/298.22 , a__isNatKind^#(plus(V1, V2)) -> 1180.43/298.22 c_61(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1180.43/298.22 , a__isNatKind^#(x(V1, V2)) -> 1180.43/298.22 c_62(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1180.43/298.22 , a__x^#(X1, X2) -> c_53(X1, X2) 1180.43/298.22 , a__x^#(N, s(M)) -> 1180.43/298.22 c_54(a__U71^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1180.43/298.22 and(isNat(N), isNatKind(N))), 1180.43/298.22 M, 1180.43/298.22 N)) 1180.43/298.22 , a__x^#(N, 0()) -> 1180.43/298.22 c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N)))) 1180.43/298.22 , a__and^#(X1, X2) -> c_56(X1, X2) 1180.43/298.22 , a__and^#(tt(), X) -> c_57(mark^#(X)) 1180.43/298.22 , a__U51^#(X1, X2, X3) -> c_44(X1, X2, X3) 1180.43/298.22 , a__U51^#(tt(), M, N) -> c_45(a__plus^#(mark(N), mark(M))) 1180.43/298.22 , a__U61^#(X) -> c_49(X) 1180.43/298.22 , a__U71^#(X1, X2, X3) -> c_51(X1, X2, X3) 1180.43/298.22 , a__U71^#(tt(), M, N) -> 1180.43/298.22 c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 1180.43/298.22 Strict Trs: 1180.43/298.22 { a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1180.43/298.22 , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2) 1180.43/298.22 , a__U12(X1, X2) -> U12(X1, X2) 1180.43/298.22 , a__U12(tt(), V2) -> a__U13(a__isNat(V2)) 1180.43/298.22 , a__isNat(X) -> isNat(X) 1180.43/298.22 , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1180.43/298.22 , a__isNat(0()) -> tt() 1180.43/298.22 , a__isNat(plus(V1, V2)) -> 1180.43/298.22 a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 1180.43/298.22 , a__isNat(x(V1, V2)) -> 1180.43/298.22 a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 1180.43/298.22 , a__U13(X) -> U13(X) 1180.43/298.22 , a__U13(tt()) -> tt() 1180.43/298.22 , a__U21(X1, X2) -> U21(X1, X2) 1180.43/298.22 , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) 1180.43/298.22 , a__U22(X) -> U22(X) 1180.43/298.22 , a__U22(tt()) -> tt() 1180.43/298.22 , a__U31(X1, X2, X3) -> U31(X1, X2, X3) 1180.43/298.22 , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2) 1180.43/298.22 , a__U32(X1, X2) -> U32(X1, X2) 1180.43/298.22 , a__U32(tt(), V2) -> a__U33(a__isNat(V2)) 1180.43/298.22 , a__U33(X) -> U33(X) 1180.43/298.22 , a__U33(tt()) -> tt() 1180.43/298.22 , a__U41(X1, X2) -> U41(X1, X2) 1180.43/298.22 , a__U41(tt(), N) -> mark(N) 1180.43/298.22 , mark(tt()) -> tt() 1180.43/298.22 , mark(s(X)) -> s(mark(X)) 1180.43/298.22 , mark(0()) -> 0() 1180.43/298.22 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1180.83/298.32 , mark(isNatKind(X)) -> a__isNatKind(X) 1180.83/298.32 , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 1180.83/298.32 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 1180.83/298.32 , mark(isNat(X)) -> a__isNat(X) 1180.83/298.32 , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1180.83/298.32 , mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1180.83/298.32 , mark(U13(X)) -> a__U13(mark(X)) 1180.83/298.32 , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1180.83/298.32 , mark(U22(X)) -> a__U22(mark(X)) 1180.83/298.32 , mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 1180.83/298.32 , mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1180.83/298.32 , mark(U33(X)) -> a__U33(mark(X)) 1180.83/298.32 , mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1180.83/298.32 , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1180.83/298.32 , mark(U61(X)) -> a__U61(mark(X)) 1180.83/298.32 , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 1180.83/298.32 , a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1180.83/298.32 , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1180.83/298.32 , a__plus(X1, X2) -> plus(X1, X2) 1180.83/298.32 , a__plus(N, s(M)) -> 1180.83/298.32 a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), 1180.83/298.32 and(isNat(N), isNatKind(N))), 1180.83/298.32 M, 1180.83/298.32 N) 1180.83/298.32 , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) 1180.83/298.32 , a__U61(X) -> U61(X) 1180.83/298.32 , a__U61(tt()) -> 0() 1180.83/298.32 , a__U71(X1, X2, X3) -> U71(X1, X2, X3) 1180.83/298.32 , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 1180.83/298.32 , a__x(X1, X2) -> x(X1, X2) 1180.83/298.32 , a__x(N, s(M)) -> 1180.83/298.32 a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), 1180.83/298.32 and(isNat(N), isNatKind(N))), 1180.83/298.32 M, 1180.83/298.32 N) 1180.83/298.32 , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(N))) 1180.83/298.32 , a__and(X1, X2) -> and(X1, X2) 1180.83/298.32 , a__and(tt(), X) -> mark(X) 1180.83/298.32 , a__isNatKind(X) -> isNatKind(X) 1180.83/298.32 , a__isNatKind(s(V1)) -> a__isNatKind(V1) 1180.83/298.32 , a__isNatKind(0()) -> tt() 1180.83/298.32 , a__isNatKind(plus(V1, V2)) -> 1180.83/298.32 a__and(a__isNatKind(V1), isNatKind(V2)) 1180.83/298.32 , a__isNatKind(x(V1, V2)) -> 1180.83/298.32 a__and(a__isNatKind(V1), isNatKind(V2)) } 1180.83/298.32 Weak DPs: 1180.83/298.32 { a__U13^#(tt()) -> c_11() 1180.83/298.32 , a__isNat^#(0()) -> c_7() 1180.83/298.32 , a__U22^#(tt()) -> c_15() 1180.83/298.32 , a__U33^#(tt()) -> c_21() 1180.83/298.32 , mark^#(tt()) -> c_24() 1180.83/298.32 , mark^#(0()) -> c_26() 1180.83/298.32 , a__isNatKind^#(0()) -> c_60() 1180.83/298.32 , a__U61^#(tt()) -> c_50() } 1180.83/298.32 Obligation: 1180.83/298.32 runtime complexity 1180.83/298.32 Answer: 1180.83/298.32 MAYBE 1180.83/298.32 1180.83/298.32 Empty strict component of the problem is NOT empty. 1180.83/298.32 1180.83/298.32 1180.83/298.32 Arrrr.. 1181.99/299.10 EOF