MAYBE 1181.01/297.40 MAYBE 1181.01/297.40 1181.01/297.40 We are left with following problem, upon which TcT provides the 1181.01/297.40 certificate MAYBE. 1181.01/297.40 1181.01/297.40 Strict Trs: 1181.01/297.40 { a__U11(X1, X2) -> U11(X1, X2) 1181.01/297.40 , a__U11(tt(), V2) -> a__U12(a__isNat(V2)) 1181.01/297.40 , a__U12(X) -> U12(X) 1181.01/297.40 , a__U12(tt()) -> tt() 1181.01/297.40 , a__isNat(X) -> isNat(X) 1181.01/297.40 , a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 1181.01/297.40 , a__isNat(0()) -> tt() 1181.01/297.40 , a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 1181.01/297.40 , a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 1181.01/297.40 , a__U21(X) -> U21(X) 1181.01/297.40 , a__U21(tt()) -> tt() 1181.01/297.40 , a__U31(X1, X2) -> U31(X1, X2) 1181.01/297.40 , a__U31(tt(), V2) -> a__U32(a__isNat(V2)) 1181.01/297.40 , a__U32(X) -> U32(X) 1181.01/297.40 , a__U32(tt()) -> tt() 1181.01/297.40 , a__U41(X1, X2) -> U41(X1, X2) 1181.01/297.40 , a__U41(tt(), N) -> mark(N) 1181.01/297.40 , mark(tt()) -> tt() 1181.01/297.40 , mark(s(X)) -> s(mark(X)) 1181.01/297.40 , mark(0()) -> 0() 1181.01/297.40 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1181.01/297.40 , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 1181.01/297.40 , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1181.01/297.40 , mark(U12(X)) -> a__U12(mark(X)) 1181.01/297.40 , mark(isNat(X)) -> a__isNat(X) 1181.01/297.40 , mark(U21(X)) -> a__U21(mark(X)) 1181.01/297.40 , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1181.01/297.40 , mark(U32(X)) -> a__U32(mark(X)) 1181.01/297.40 , mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1181.01/297.40 , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1181.01/297.40 , mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 1181.01/297.40 , mark(U61(X)) -> a__U61(mark(X)) 1181.01/297.40 , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 1181.01/297.40 , mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 1181.01/297.40 , a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1181.01/297.40 , a__U51(tt(), M, N) -> a__U52(a__isNat(N), M, N) 1181.01/297.40 , a__U52(X1, X2, X3) -> U52(X1, X2, X3) 1181.01/297.40 , a__U52(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1181.01/297.40 , a__plus(X1, X2) -> plus(X1, X2) 1181.01/297.40 , a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 1181.01/297.40 , a__plus(N, 0()) -> a__U41(a__isNat(N), N) 1181.01/297.40 , a__U61(X) -> U61(X) 1181.01/297.40 , a__U61(tt()) -> 0() 1181.01/297.40 , a__U71(X1, X2, X3) -> U71(X1, X2, X3) 1181.01/297.40 , a__U71(tt(), M, N) -> a__U72(a__isNat(N), M, N) 1181.01/297.40 , a__U72(X1, X2, X3) -> U72(X1, X2, X3) 1181.01/297.40 , a__U72(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 1181.01/297.40 , a__x(X1, X2) -> x(X1, X2) 1181.01/297.40 , a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 1181.01/297.40 , a__x(N, 0()) -> a__U61(a__isNat(N)) } 1181.01/297.40 Obligation: 1181.01/297.40 runtime complexity 1181.01/297.40 Answer: 1181.01/297.40 MAYBE 1181.01/297.40 1181.01/297.40 None of the processors succeeded. 1181.01/297.40 1181.01/297.40 Details of failed attempt(s): 1181.01/297.40 ----------------------------- 1181.01/297.40 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 1181.01/297.40 following reason: 1181.01/297.40 1181.01/297.40 Computation stopped due to timeout after 297.0 seconds. 1181.01/297.40 1181.01/297.40 2) 'Best' failed due to the following reason: 1181.01/297.40 1181.01/297.40 None of the processors succeeded. 1181.01/297.40 1181.01/297.40 Details of failed attempt(s): 1181.01/297.40 ----------------------------- 1181.01/297.40 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 1181.01/297.40 seconds)' failed due to the following reason: 1181.01/297.40 1181.01/297.40 Computation stopped due to timeout after 148.0 seconds. 1181.01/297.40 1181.01/297.40 2) 'Best' failed due to the following reason: 1181.01/297.40 1181.01/297.40 None of the processors succeeded. 1181.01/297.40 1181.01/297.40 Details of failed attempt(s): 1181.01/297.40 ----------------------------- 1181.01/297.40 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 1181.01/297.40 following reason: 1181.01/297.40 1181.01/297.40 The processor is inapplicable, reason: 1181.01/297.40 Processor only applicable for innermost runtime complexity analysis 1181.01/297.40 1181.01/297.40 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 1181.01/297.40 to the following reason: 1181.01/297.40 1181.01/297.40 The processor is inapplicable, reason: 1181.01/297.40 Processor only applicable for innermost runtime complexity analysis 1181.01/297.40 1181.01/297.40 1181.01/297.40 3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 1181.01/297.40 failed due to the following reason: 1181.01/297.40 1181.01/297.40 None of the processors succeeded. 1181.01/297.40 1181.01/297.40 Details of failed attempt(s): 1181.01/297.40 ----------------------------- 1181.01/297.40 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 1181.01/297.40 failed due to the following reason: 1181.01/297.40 1181.01/297.40 match-boundness of the problem could not be verified. 1181.01/297.40 1181.01/297.40 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 1181.01/297.40 failed due to the following reason: 1181.01/297.40 1181.01/297.40 match-boundness of the problem could not be verified. 1181.01/297.40 1181.01/297.40 1181.01/297.40 1181.01/297.40 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 1181.01/297.40 the following reason: 1181.01/297.40 1181.01/297.40 We add the following weak dependency pairs: 1181.01/297.40 1181.01/297.40 Strict DPs: 1181.01/297.40 { a__U11^#(X1, X2) -> c_1(X1, X2) 1181.01/297.40 , a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2))) 1181.01/297.40 , a__U12^#(X) -> c_3(X) 1181.01/297.40 , a__U12^#(tt()) -> c_4() 1181.01/297.40 , a__isNat^#(X) -> c_5(X) 1181.01/297.40 , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1))) 1181.01/297.40 , a__isNat^#(0()) -> c_7() 1181.01/297.40 , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2)) 1181.01/297.40 , a__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2)) 1181.01/297.40 , a__U21^#(X) -> c_10(X) 1181.01/297.40 , a__U21^#(tt()) -> c_11() 1181.01/297.40 , a__U31^#(X1, X2) -> c_12(X1, X2) 1181.01/297.40 , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2))) 1181.01/297.40 , a__U32^#(X) -> c_14(X) 1181.01/297.40 , a__U32^#(tt()) -> c_15() 1181.01/297.40 , a__U41^#(X1, X2) -> c_16(X1, X2) 1181.01/297.40 , a__U41^#(tt(), N) -> c_17(mark^#(N)) 1181.01/297.40 , mark^#(tt()) -> c_18() 1181.01/297.40 , mark^#(s(X)) -> c_19(mark^#(X)) 1181.01/297.40 , mark^#(0()) -> c_20() 1181.01/297.40 , mark^#(plus(X1, X2)) -> c_21(a__plus^#(mark(X1), mark(X2))) 1181.01/297.40 , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2))) 1181.01/297.40 , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2)) 1181.01/297.40 , mark^#(U12(X)) -> c_24(a__U12^#(mark(X))) 1181.01/297.40 , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) 1181.01/297.40 , mark^#(U21(X)) -> c_26(a__U21^#(mark(X))) 1181.01/297.40 , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2)) 1181.01/297.40 , mark^#(U32(X)) -> c_28(a__U32^#(mark(X))) 1181.01/297.40 , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2)) 1181.01/297.40 , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3)) 1181.01/297.40 , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3)) 1181.01/297.40 , mark^#(U61(X)) -> c_32(a__U61^#(mark(X))) 1181.01/297.40 , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3)) 1181.01/297.40 , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3)) 1181.01/297.40 , a__plus^#(X1, X2) -> c_39(X1, X2) 1181.01/297.40 , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N)) 1181.01/297.40 , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N)) 1181.01/297.40 , a__x^#(X1, X2) -> c_48(X1, X2) 1181.01/297.40 , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N)) 1181.01/297.40 , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N))) 1181.01/297.40 , a__U51^#(X1, X2, X3) -> c_35(X1, X2, X3) 1181.01/297.40 , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N)) 1181.01/297.40 , a__U52^#(X1, X2, X3) -> c_37(X1, X2, X3) 1181.01/297.40 , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M))) 1181.01/297.40 , a__U61^#(X) -> c_42(X) 1181.01/297.40 , a__U61^#(tt()) -> c_43() 1181.01/297.40 , a__U71^#(X1, X2, X3) -> c_44(X1, X2, X3) 1181.01/297.40 , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N)) 1181.01/297.40 , a__U72^#(X1, X2, X3) -> c_46(X1, X2, X3) 1181.01/297.40 , a__U72^#(tt(), M, N) -> 1181.01/297.40 c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 1181.01/297.40 1181.01/297.40 and mark the set of starting terms. 1181.01/297.40 1181.01/297.40 We are left with following problem, upon which TcT provides the 1181.01/297.40 certificate MAYBE. 1181.01/297.40 1181.01/297.40 Strict DPs: 1181.01/297.40 { a__U11^#(X1, X2) -> c_1(X1, X2) 1181.01/297.40 , a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2))) 1181.01/297.40 , a__U12^#(X) -> c_3(X) 1181.01/297.40 , a__U12^#(tt()) -> c_4() 1181.01/297.40 , a__isNat^#(X) -> c_5(X) 1181.01/297.40 , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1))) 1181.01/297.40 , a__isNat^#(0()) -> c_7() 1181.01/297.40 , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2)) 1181.01/297.40 , a__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2)) 1181.01/297.40 , a__U21^#(X) -> c_10(X) 1181.01/297.40 , a__U21^#(tt()) -> c_11() 1181.01/297.40 , a__U31^#(X1, X2) -> c_12(X1, X2) 1181.01/297.40 , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2))) 1181.01/297.40 , a__U32^#(X) -> c_14(X) 1181.01/297.40 , a__U32^#(tt()) -> c_15() 1181.01/297.40 , a__U41^#(X1, X2) -> c_16(X1, X2) 1181.01/297.40 , a__U41^#(tt(), N) -> c_17(mark^#(N)) 1181.01/297.40 , mark^#(tt()) -> c_18() 1181.01/297.40 , mark^#(s(X)) -> c_19(mark^#(X)) 1181.01/297.40 , mark^#(0()) -> c_20() 1181.01/297.40 , mark^#(plus(X1, X2)) -> c_21(a__plus^#(mark(X1), mark(X2))) 1181.01/297.40 , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2))) 1181.01/297.40 , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2)) 1181.01/297.40 , mark^#(U12(X)) -> c_24(a__U12^#(mark(X))) 1181.01/297.40 , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) 1181.01/297.40 , mark^#(U21(X)) -> c_26(a__U21^#(mark(X))) 1181.01/297.40 , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2)) 1181.01/297.40 , mark^#(U32(X)) -> c_28(a__U32^#(mark(X))) 1181.01/297.40 , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2)) 1181.01/297.41 , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3)) 1181.01/297.41 , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3)) 1181.01/297.41 , mark^#(U61(X)) -> c_32(a__U61^#(mark(X))) 1181.01/297.41 , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3)) 1181.01/297.41 , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3)) 1181.01/297.41 , a__plus^#(X1, X2) -> c_39(X1, X2) 1181.01/297.41 , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N)) 1181.01/297.41 , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N)) 1181.01/297.41 , a__x^#(X1, X2) -> c_48(X1, X2) 1181.01/297.41 , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N)) 1181.01/297.41 , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N))) 1181.01/297.41 , a__U51^#(X1, X2, X3) -> c_35(X1, X2, X3) 1181.01/297.41 , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N)) 1181.01/297.41 , a__U52^#(X1, X2, X3) -> c_37(X1, X2, X3) 1181.01/297.41 , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M))) 1181.01/297.41 , a__U61^#(X) -> c_42(X) 1181.01/297.41 , a__U61^#(tt()) -> c_43() 1181.01/297.41 , a__U71^#(X1, X2, X3) -> c_44(X1, X2, X3) 1181.01/297.41 , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N)) 1181.01/297.41 , a__U72^#(X1, X2, X3) -> c_46(X1, X2, X3) 1181.01/297.41 , a__U72^#(tt(), M, N) -> 1181.01/297.41 c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 1181.01/297.41 Strict Trs: 1181.01/297.41 { a__U11(X1, X2) -> U11(X1, X2) 1181.01/297.41 , a__U11(tt(), V2) -> a__U12(a__isNat(V2)) 1181.01/297.41 , a__U12(X) -> U12(X) 1181.01/297.41 , a__U12(tt()) -> tt() 1181.01/297.41 , a__isNat(X) -> isNat(X) 1181.01/297.41 , a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 1181.01/297.41 , a__isNat(0()) -> tt() 1181.01/297.41 , a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 1181.01/297.41 , a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 1181.01/297.41 , a__U21(X) -> U21(X) 1181.01/297.41 , a__U21(tt()) -> tt() 1181.01/297.41 , a__U31(X1, X2) -> U31(X1, X2) 1181.01/297.41 , a__U31(tt(), V2) -> a__U32(a__isNat(V2)) 1181.01/297.41 , a__U32(X) -> U32(X) 1181.01/297.41 , a__U32(tt()) -> tt() 1181.01/297.41 , a__U41(X1, X2) -> U41(X1, X2) 1181.01/297.41 , a__U41(tt(), N) -> mark(N) 1181.01/297.41 , mark(tt()) -> tt() 1181.01/297.41 , mark(s(X)) -> s(mark(X)) 1181.01/297.41 , mark(0()) -> 0() 1181.01/297.41 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1181.01/297.41 , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 1181.01/297.41 , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1181.01/297.41 , mark(U12(X)) -> a__U12(mark(X)) 1181.01/297.41 , mark(isNat(X)) -> a__isNat(X) 1181.01/297.41 , mark(U21(X)) -> a__U21(mark(X)) 1181.01/297.41 , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1181.01/297.41 , mark(U32(X)) -> a__U32(mark(X)) 1181.01/297.41 , mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1181.01/297.41 , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1181.01/297.41 , mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 1181.01/297.41 , mark(U61(X)) -> a__U61(mark(X)) 1181.01/297.41 , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 1181.01/297.41 , mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 1181.01/297.41 , a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1181.01/297.41 , a__U51(tt(), M, N) -> a__U52(a__isNat(N), M, N) 1181.01/297.41 , a__U52(X1, X2, X3) -> U52(X1, X2, X3) 1181.01/297.41 , a__U52(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1181.01/297.41 , a__plus(X1, X2) -> plus(X1, X2) 1181.01/297.41 , a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 1181.01/297.41 , a__plus(N, 0()) -> a__U41(a__isNat(N), N) 1181.01/297.41 , a__U61(X) -> U61(X) 1181.01/297.41 , a__U61(tt()) -> 0() 1181.01/297.41 , a__U71(X1, X2, X3) -> U71(X1, X2, X3) 1181.01/297.41 , a__U71(tt(), M, N) -> a__U72(a__isNat(N), M, N) 1181.01/297.41 , a__U72(X1, X2, X3) -> U72(X1, X2, X3) 1181.01/297.41 , a__U72(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 1181.01/297.41 , a__x(X1, X2) -> x(X1, X2) 1181.01/297.41 , a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 1181.01/297.41 , a__x(N, 0()) -> a__U61(a__isNat(N)) } 1181.01/297.41 Obligation: 1181.01/297.41 runtime complexity 1181.01/297.41 Answer: 1181.01/297.41 MAYBE 1181.01/297.41 1181.01/297.41 We estimate the number of application of {4,7,11,15,18,20,46} by 1181.01/297.41 applications of Pre({4,7,11,15,18,20,46}) = 1181.01/297.41 {1,2,3,5,6,10,12,13,14,16,17,19,24,25,26,28,32,35,38,40,41,43,45,47,49}. 1181.01/297.41 Here rules are labeled as follows: 1181.01/297.41 1181.01/297.41 DPs: 1181.01/297.41 { 1: a__U11^#(X1, X2) -> c_1(X1, X2) 1181.01/297.41 , 2: a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2))) 1181.01/297.41 , 3: a__U12^#(X) -> c_3(X) 1181.01/297.41 , 4: a__U12^#(tt()) -> c_4() 1181.01/297.41 , 5: a__isNat^#(X) -> c_5(X) 1181.01/297.41 , 6: a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1))) 1181.01/297.41 , 7: a__isNat^#(0()) -> c_7() 1181.01/297.41 , 8: a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2)) 1181.01/297.41 , 9: a__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2)) 1181.01/297.41 , 10: a__U21^#(X) -> c_10(X) 1181.01/297.41 , 11: a__U21^#(tt()) -> c_11() 1181.01/297.41 , 12: a__U31^#(X1, X2) -> c_12(X1, X2) 1181.01/297.41 , 13: a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2))) 1181.01/297.41 , 14: a__U32^#(X) -> c_14(X) 1181.01/297.41 , 15: a__U32^#(tt()) -> c_15() 1181.01/297.41 , 16: a__U41^#(X1, X2) -> c_16(X1, X2) 1181.01/297.41 , 17: a__U41^#(tt(), N) -> c_17(mark^#(N)) 1181.01/297.41 , 18: mark^#(tt()) -> c_18() 1181.01/297.41 , 19: mark^#(s(X)) -> c_19(mark^#(X)) 1181.01/297.41 , 20: mark^#(0()) -> c_20() 1181.01/297.41 , 21: mark^#(plus(X1, X2)) -> c_21(a__plus^#(mark(X1), mark(X2))) 1181.01/297.41 , 22: mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2))) 1181.01/297.41 , 23: mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2)) 1181.01/297.41 , 24: mark^#(U12(X)) -> c_24(a__U12^#(mark(X))) 1181.01/297.41 , 25: mark^#(isNat(X)) -> c_25(a__isNat^#(X)) 1181.01/297.41 , 26: mark^#(U21(X)) -> c_26(a__U21^#(mark(X))) 1181.01/297.41 , 27: mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2)) 1181.01/297.41 , 28: mark^#(U32(X)) -> c_28(a__U32^#(mark(X))) 1181.01/297.41 , 29: mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2)) 1181.01/297.41 , 30: mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3)) 1181.01/297.41 , 31: mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3)) 1181.01/297.41 , 32: mark^#(U61(X)) -> c_32(a__U61^#(mark(X))) 1181.01/297.41 , 33: mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3)) 1181.01/297.41 , 34: mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3)) 1181.01/297.41 , 35: a__plus^#(X1, X2) -> c_39(X1, X2) 1181.01/297.41 , 36: a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N)) 1181.01/297.41 , 37: a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N)) 1181.01/297.41 , 38: a__x^#(X1, X2) -> c_48(X1, X2) 1181.01/297.41 , 39: a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N)) 1181.01/297.41 , 40: a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N))) 1181.01/297.41 , 41: a__U51^#(X1, X2, X3) -> c_35(X1, X2, X3) 1181.01/297.41 , 42: a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N)) 1181.01/297.41 , 43: a__U52^#(X1, X2, X3) -> c_37(X1, X2, X3) 1181.01/297.41 , 44: a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M))) 1181.01/297.41 , 45: a__U61^#(X) -> c_42(X) 1181.01/297.41 , 46: a__U61^#(tt()) -> c_43() 1181.01/297.41 , 47: a__U71^#(X1, X2, X3) -> c_44(X1, X2, X3) 1181.01/297.41 , 48: a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N)) 1181.01/297.41 , 49: a__U72^#(X1, X2, X3) -> c_46(X1, X2, X3) 1181.01/297.41 , 50: a__U72^#(tt(), M, N) -> 1181.01/297.41 c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 1181.01/297.41 1181.01/297.41 We are left with following problem, upon which TcT provides the 1181.01/297.41 certificate MAYBE. 1181.01/297.41 1181.01/297.41 Strict DPs: 1181.01/297.41 { a__U11^#(X1, X2) -> c_1(X1, X2) 1181.01/297.41 , a__U11^#(tt(), V2) -> c_2(a__U12^#(a__isNat(V2))) 1181.01/297.41 , a__U12^#(X) -> c_3(X) 1181.01/297.41 , a__isNat^#(X) -> c_5(X) 1181.01/297.41 , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNat(V1))) 1181.01/297.41 , a__isNat^#(plus(V1, V2)) -> c_8(a__U11^#(a__isNat(V1), V2)) 1181.01/297.41 , a__isNat^#(x(V1, V2)) -> c_9(a__U31^#(a__isNat(V1), V2)) 1181.01/297.41 , a__U21^#(X) -> c_10(X) 1181.01/297.41 , a__U31^#(X1, X2) -> c_12(X1, X2) 1181.01/297.41 , a__U31^#(tt(), V2) -> c_13(a__U32^#(a__isNat(V2))) 1181.01/297.41 , a__U32^#(X) -> c_14(X) 1181.01/297.41 , a__U41^#(X1, X2) -> c_16(X1, X2) 1181.01/297.41 , a__U41^#(tt(), N) -> c_17(mark^#(N)) 1181.01/297.41 , mark^#(s(X)) -> c_19(mark^#(X)) 1181.01/297.41 , mark^#(plus(X1, X2)) -> c_21(a__plus^#(mark(X1), mark(X2))) 1181.01/297.41 , mark^#(x(X1, X2)) -> c_22(a__x^#(mark(X1), mark(X2))) 1181.01/297.41 , mark^#(U11(X1, X2)) -> c_23(a__U11^#(mark(X1), X2)) 1181.01/297.41 , mark^#(U12(X)) -> c_24(a__U12^#(mark(X))) 1181.01/297.41 , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) 1181.01/297.41 , mark^#(U21(X)) -> c_26(a__U21^#(mark(X))) 1181.01/297.41 , mark^#(U31(X1, X2)) -> c_27(a__U31^#(mark(X1), X2)) 1181.01/297.41 , mark^#(U32(X)) -> c_28(a__U32^#(mark(X))) 1181.01/297.41 , mark^#(U41(X1, X2)) -> c_29(a__U41^#(mark(X1), X2)) 1181.01/297.41 , mark^#(U51(X1, X2, X3)) -> c_30(a__U51^#(mark(X1), X2, X3)) 1181.01/297.41 , mark^#(U52(X1, X2, X3)) -> c_31(a__U52^#(mark(X1), X2, X3)) 1181.01/297.41 , mark^#(U61(X)) -> c_32(a__U61^#(mark(X))) 1181.01/297.41 , mark^#(U71(X1, X2, X3)) -> c_33(a__U71^#(mark(X1), X2, X3)) 1181.01/297.41 , mark^#(U72(X1, X2, X3)) -> c_34(a__U72^#(mark(X1), X2, X3)) 1181.01/297.41 , a__plus^#(X1, X2) -> c_39(X1, X2) 1181.01/297.41 , a__plus^#(N, s(M)) -> c_40(a__U51^#(a__isNat(M), M, N)) 1181.01/297.41 , a__plus^#(N, 0()) -> c_41(a__U41^#(a__isNat(N), N)) 1181.01/297.47 , a__x^#(X1, X2) -> c_48(X1, X2) 1181.01/297.47 , a__x^#(N, s(M)) -> c_49(a__U71^#(a__isNat(M), M, N)) 1181.01/297.47 , a__x^#(N, 0()) -> c_50(a__U61^#(a__isNat(N))) 1181.01/297.47 , a__U51^#(X1, X2, X3) -> c_35(X1, X2, X3) 1181.01/297.47 , a__U51^#(tt(), M, N) -> c_36(a__U52^#(a__isNat(N), M, N)) 1181.01/297.47 , a__U52^#(X1, X2, X3) -> c_37(X1, X2, X3) 1181.01/297.47 , a__U52^#(tt(), M, N) -> c_38(a__plus^#(mark(N), mark(M))) 1181.01/297.47 , a__U61^#(X) -> c_42(X) 1181.01/297.47 , a__U71^#(X1, X2, X3) -> c_44(X1, X2, X3) 1181.01/297.47 , a__U71^#(tt(), M, N) -> c_45(a__U72^#(a__isNat(N), M, N)) 1181.01/297.47 , a__U72^#(X1, X2, X3) -> c_46(X1, X2, X3) 1181.01/297.47 , a__U72^#(tt(), M, N) -> 1181.01/297.47 c_47(a__plus^#(a__x(mark(N), mark(M)), mark(N))) } 1181.01/297.47 Strict Trs: 1181.01/297.47 { a__U11(X1, X2) -> U11(X1, X2) 1181.01/297.47 , a__U11(tt(), V2) -> a__U12(a__isNat(V2)) 1181.01/297.47 , a__U12(X) -> U12(X) 1181.01/297.47 , a__U12(tt()) -> tt() 1181.01/297.47 , a__isNat(X) -> isNat(X) 1181.01/297.47 , a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 1181.01/297.47 , a__isNat(0()) -> tt() 1181.01/297.47 , a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 1181.01/297.47 , a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 1181.01/297.47 , a__U21(X) -> U21(X) 1181.01/297.47 , a__U21(tt()) -> tt() 1181.01/297.47 , a__U31(X1, X2) -> U31(X1, X2) 1181.01/297.47 , a__U31(tt(), V2) -> a__U32(a__isNat(V2)) 1181.01/297.47 , a__U32(X) -> U32(X) 1181.01/297.47 , a__U32(tt()) -> tt() 1181.01/297.47 , a__U41(X1, X2) -> U41(X1, X2) 1181.01/297.47 , a__U41(tt(), N) -> mark(N) 1181.01/297.47 , mark(tt()) -> tt() 1181.01/297.47 , mark(s(X)) -> s(mark(X)) 1181.01/297.47 , mark(0()) -> 0() 1181.01/297.47 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1181.01/297.47 , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 1181.01/297.47 , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1181.01/297.47 , mark(U12(X)) -> a__U12(mark(X)) 1181.01/297.47 , mark(isNat(X)) -> a__isNat(X) 1181.01/297.47 , mark(U21(X)) -> a__U21(mark(X)) 1181.01/297.47 , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1181.01/297.47 , mark(U32(X)) -> a__U32(mark(X)) 1181.01/297.47 , mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 1181.01/297.47 , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1181.01/297.47 , mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 1181.01/297.47 , mark(U61(X)) -> a__U61(mark(X)) 1181.01/297.47 , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 1181.01/297.47 , mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 1181.01/297.48 , a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1181.01/297.48 , a__U51(tt(), M, N) -> a__U52(a__isNat(N), M, N) 1181.01/297.48 , a__U52(X1, X2, X3) -> U52(X1, X2, X3) 1181.01/297.48 , a__U52(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1181.01/297.48 , a__plus(X1, X2) -> plus(X1, X2) 1181.01/297.48 , a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 1181.01/297.48 , a__plus(N, 0()) -> a__U41(a__isNat(N), N) 1181.01/297.48 , a__U61(X) -> U61(X) 1181.01/297.48 , a__U61(tt()) -> 0() 1181.01/297.48 , a__U71(X1, X2, X3) -> U71(X1, X2, X3) 1181.01/297.48 , a__U71(tt(), M, N) -> a__U72(a__isNat(N), M, N) 1181.01/297.48 , a__U72(X1, X2, X3) -> U72(X1, X2, X3) 1181.01/297.48 , a__U72(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 1181.01/297.48 , a__x(X1, X2) -> x(X1, X2) 1181.01/297.48 , a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 1181.01/297.48 , a__x(N, 0()) -> a__U61(a__isNat(N)) } 1181.01/297.48 Weak DPs: 1181.01/297.48 { a__U12^#(tt()) -> c_4() 1181.01/297.48 , a__isNat^#(0()) -> c_7() 1181.01/297.48 , a__U21^#(tt()) -> c_11() 1181.01/297.48 , a__U32^#(tt()) -> c_15() 1181.01/297.48 , mark^#(tt()) -> c_18() 1181.01/297.48 , mark^#(0()) -> c_20() 1181.01/297.48 , a__U61^#(tt()) -> c_43() } 1181.01/297.48 Obligation: 1181.01/297.48 runtime complexity 1181.01/297.48 Answer: 1181.01/297.48 MAYBE 1181.01/297.48 1181.01/297.48 Empty strict component of the problem is NOT empty. 1181.01/297.48 1181.01/297.48 1181.01/297.48 Arrrr.. 1182.22/298.42 EOF