/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 859 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, V1)) -> mark(U12(isNatList(V1))) active(U12(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isNatList(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) active(U52(tt, V2)) -> mark(U53(isNatList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(isNatIListKind(nil)) -> mark(tt) active(isNatIListKind(zeros)) -> mark(tt) active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) active(isNatKind(0)) -> mark(tt) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_nil -> nil encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, V1)) -> mark(U12(isNatList(V1))) active(U12(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isNatList(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) active(U52(tt, V2)) -> mark(U53(isNatList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(isNatIListKind(nil)) -> mark(tt) active(isNatIListKind(zeros)) -> mark(tt) active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) active(isNatKind(0)) -> mark(tt) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_nil -> nil encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, V1)) -> mark(U12(isNatList(V1))) active(U12(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isNatList(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) active(U52(tt, V2)) -> mark(U53(isNatList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(isNatIListKind(nil)) -> mark(tt) active(isNatIListKind(zeros)) -> mark(tt) active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) active(isNatKind(0)) -> mark(tt) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_nil -> nil encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, V1)) -> mark(U12(isNatList(V1))) active(U12(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isNatList(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) active(U52(tt, V2)) -> mark(U53(isNatList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(isNatIListKind(nil)) -> mark(tt) active(isNatIListKind(zeros)) -> mark(tt) active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) active(isNatKind(0)) -> mark(tt) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_nil -> nil encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence U52(ok(X1), ok(X2)) ->^+ ok(U52(X1, X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, V1)) -> mark(U12(isNatList(V1))) active(U12(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isNatList(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) active(U52(tt, V2)) -> mark(U53(isNatList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(isNatIListKind(nil)) -> mark(tt) active(isNatIListKind(zeros)) -> mark(tt) active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) active(isNatKind(0)) -> mark(tt) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_nil -> nil encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, V1)) -> mark(U12(isNatList(V1))) active(U12(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isNatList(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) active(U52(tt, V2)) -> mark(U53(isNatList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(isNatIListKind(nil)) -> mark(tt) active(isNatIListKind(zeros)) -> mark(tt) active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) active(isNatKind(0)) -> mark(tt) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_nil -> nil encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST