/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 1916 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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0)) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0, XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0) -> ok(0) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 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(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3, x_4)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U82(x_1, x_2)) -> U82(encArg(x_1), encArg(x_2)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatural(x_1)) -> isNatural(encArg(x_1)) encArg(cons_isLNat(x_1)) -> isLNat(encArg(x_1)) encArg(cons_isPLNat(x_1)) -> isPLNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_snd(x_1) -> snd(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_head(x_1) -> head(encArg(x_1)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U81(x_1, x_2, x_3, x_4) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U82(x_1, x_2) -> U82(encArg(x_1), encArg(x_2)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatural(x_1) -> isNatural(encArg(x_1)) encode_isLNat(x_1) -> isLNat(encArg(x_1)) encode_isPLNat(x_1) -> isPLNat(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) 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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0)) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0, XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0) -> ok(0) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3, x_4)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U82(x_1, x_2)) -> U82(encArg(x_1), encArg(x_2)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatural(x_1)) -> isNatural(encArg(x_1)) encArg(cons_isLNat(x_1)) -> isLNat(encArg(x_1)) encArg(cons_isPLNat(x_1)) -> isPLNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_snd(x_1) -> snd(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_head(x_1) -> head(encArg(x_1)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U81(x_1, x_2, x_3, x_4) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U82(x_1, x_2) -> U82(encArg(x_1), encArg(x_2)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatural(x_1) -> isNatural(encArg(x_1)) encode_isLNat(x_1) -> isLNat(encArg(x_1)) encode_isPLNat(x_1) -> isPLNat(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) 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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0)) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0, XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0) -> ok(0) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3, x_4)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U82(x_1, x_2)) -> U82(encArg(x_1), encArg(x_2)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatural(x_1)) -> isNatural(encArg(x_1)) encArg(cons_isLNat(x_1)) -> isLNat(encArg(x_1)) encArg(cons_isPLNat(x_1)) -> isPLNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_snd(x_1) -> snd(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_head(x_1) -> head(encArg(x_1)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U81(x_1, x_2, x_3, x_4) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U82(x_1, x_2) -> U82(encArg(x_1), encArg(x_2)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatural(x_1) -> isNatural(encArg(x_1)) encode_isLNat(x_1) -> isLNat(encArg(x_1)) encode_isPLNat(x_1) -> isPLNat(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) 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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0)) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0, XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0) -> ok(0) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3, x_4)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U82(x_1, x_2)) -> U82(encArg(x_1), encArg(x_2)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatural(x_1)) -> isNatural(encArg(x_1)) encArg(cons_isLNat(x_1)) -> isLNat(encArg(x_1)) encArg(cons_isPLNat(x_1)) -> isPLNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_snd(x_1) -> snd(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_head(x_1) -> head(encArg(x_1)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U81(x_1, x_2, x_3, x_4) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U82(x_1, x_2) -> U82(encArg(x_1), encArg(x_2)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatural(x_1) -> isNatural(encArg(x_1)) encode_isLNat(x_1) -> isLNat(encArg(x_1)) encode_isPLNat(x_1) -> isPLNat(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) 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 take(ok(X1), ok(X2)) ->^+ ok(take(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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0)) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0, XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0) -> ok(0) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3, x_4)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U82(x_1, x_2)) -> U82(encArg(x_1), encArg(x_2)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatural(x_1)) -> isNatural(encArg(x_1)) encArg(cons_isLNat(x_1)) -> isLNat(encArg(x_1)) encArg(cons_isPLNat(x_1)) -> isPLNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_snd(x_1) -> snd(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_head(x_1) -> head(encArg(x_1)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U81(x_1, x_2, x_3, x_4) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U82(x_1, x_2) -> U82(encArg(x_1), encArg(x_2)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatural(x_1) -> isNatural(encArg(x_1)) encode_isLNat(x_1) -> isLNat(encArg(x_1)) encode_isPLNat(x_1) -> isPLNat(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) 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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0)) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0, XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0) -> ok(0) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3, x_4)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U82(x_1, x_2)) -> U82(encArg(x_1), encArg(x_2)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatural(x_1)) -> isNatural(encArg(x_1)) encArg(cons_isLNat(x_1)) -> isLNat(encArg(x_1)) encArg(cons_isPLNat(x_1)) -> isPLNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_snd(x_1) -> snd(encArg(x_1)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_head(x_1) -> head(encArg(x_1)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U81(x_1, x_2, x_3, x_4) -> U81(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U82(x_1, x_2) -> U82(encArg(x_1), encArg(x_2)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNatural(x_1) -> isNatural(encArg(x_1)) encode_isLNat(x_1) -> isLNat(encArg(x_1)) encode_isPLNat(x_1) -> isPLNat(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) 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