/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 1018 ms] (6) BOUNDS(1, n^1) (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (8) TRS for Loop Detection (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (10) BEST (11) proven lower bound (12) LowerBoundPropagationProof [FINISHED, 0 ms] (13) BOUNDS(n^1, INF) (14) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 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: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 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)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 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(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(sel(X1, X2)) -> sel(proper(X1), proper(X2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 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(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) 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: FULL ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 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(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) 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: FULL ---------------------------------------- (5) CpxTrsMatchBoundsTAProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] transitions: mark0(0) -> 0 tt0() -> 0 ok0(0) -> 0 nil0() -> 0 00() -> 0 active0(0) -> 0 U1010(0, 0, 0) -> 1 fst0(0) -> 2 splitAt0(0, 0) -> 3 U110(0, 0, 0) -> 4 snd0(0) -> 5 U210(0, 0) -> 6 U310(0, 0) -> 7 U410(0, 0) -> 8 cons0(0, 0) -> 9 natsFrom0(0) -> 10 s0(0) -> 11 U510(0, 0, 0) -> 12 head0(0) -> 13 afterNth0(0, 0) -> 14 U610(0, 0) -> 15 U710(0, 0) -> 16 pair0(0, 0) -> 17 U810(0, 0, 0, 0) -> 18 U820(0, 0) -> 19 U910(0, 0) -> 20 and0(0, 0) -> 21 tail0(0) -> 22 take0(0, 0) -> 23 sel0(0, 0) -> 24 proper0(0) -> 25 isNatural0(0) -> 26 isLNat0(0) -> 27 isPLNat0(0) -> 28 top0(0) -> 29 U1011(0, 0, 0) -> 30 mark1(30) -> 1 fst1(0) -> 31 mark1(31) -> 2 splitAt1(0, 0) -> 32 mark1(32) -> 3 U111(0, 0, 0) -> 33 mark1(33) -> 4 snd1(0) -> 34 mark1(34) -> 5 U211(0, 0) -> 35 mark1(35) -> 6 U311(0, 0) -> 36 mark1(36) -> 7 U411(0, 0) -> 37 mark1(37) -> 8 cons1(0, 0) -> 38 mark1(38) -> 9 natsFrom1(0) -> 39 mark1(39) -> 10 s1(0) -> 40 mark1(40) -> 11 U511(0, 0, 0) -> 41 mark1(41) -> 12 head1(0) -> 42 mark1(42) -> 13 afterNth1(0, 0) -> 43 mark1(43) -> 14 U611(0, 0) -> 44 mark1(44) -> 15 U711(0, 0) -> 45 mark1(45) -> 16 pair1(0, 0) -> 46 mark1(46) -> 17 U811(0, 0, 0, 0) -> 47 mark1(47) -> 18 U821(0, 0) -> 48 mark1(48) -> 19 U911(0, 0) -> 49 mark1(49) -> 20 and1(0, 0) -> 50 mark1(50) -> 21 tail1(0) -> 51 mark1(51) -> 22 take1(0, 0) -> 52 mark1(52) -> 23 sel1(0, 0) -> 53 mark1(53) -> 24 tt1() -> 54 ok1(54) -> 25 nil1() -> 55 ok1(55) -> 25 01() -> 56 ok1(56) -> 25 U1011(0, 0, 0) -> 57 ok1(57) -> 1 fst1(0) -> 58 ok1(58) -> 2 splitAt1(0, 0) -> 59 ok1(59) -> 3 U111(0, 0, 0) -> 60 ok1(60) -> 4 snd1(0) -> 61 ok1(61) -> 5 U211(0, 0) -> 62 ok1(62) -> 6 U311(0, 0) -> 63 ok1(63) -> 7 U411(0, 0) -> 64 ok1(64) -> 8 cons1(0, 0) -> 65 ok1(65) -> 9 natsFrom1(0) -> 66 ok1(66) -> 10 s1(0) -> 67 ok1(67) -> 11 U511(0, 0, 0) -> 68 ok1(68) -> 12 head1(0) -> 69 ok1(69) -> 13 afterNth1(0, 0) -> 70 ok1(70) -> 14 U611(0, 0) -> 71 ok1(71) -> 15 U711(0, 0) -> 72 ok1(72) -> 16 pair1(0, 0) -> 73 ok1(73) -> 17 U811(0, 0, 0, 0) -> 74 ok1(74) -> 18 U821(0, 0) -> 75 ok1(75) -> 19 U911(0, 0) -> 76 ok1(76) -> 20 and1(0, 0) -> 77 ok1(77) -> 21 isNatural1(0) -> 78 ok1(78) -> 26 isLNat1(0) -> 79 ok1(79) -> 27 isPLNat1(0) -> 80 ok1(80) -> 28 tail1(0) -> 81 ok1(81) -> 22 take1(0, 0) -> 82 ok1(82) -> 23 sel1(0, 0) -> 83 ok1(83) -> 24 proper1(0) -> 84 top1(84) -> 29 active1(0) -> 85 top1(85) -> 29 mark1(30) -> 30 mark1(30) -> 57 mark1(31) -> 31 mark1(31) -> 58 mark1(32) -> 32 mark1(32) -> 59 mark1(33) -> 33 mark1(33) -> 60 mark1(34) -> 34 mark1(34) -> 61 mark1(35) -> 35 mark1(35) -> 62 mark1(36) -> 36 mark1(36) -> 63 mark1(37) -> 37 mark1(37) -> 64 mark1(38) -> 38 mark1(38) -> 65 mark1(39) -> 39 mark1(39) -> 66 mark1(40) -> 40 mark1(40) -> 67 mark1(41) -> 41 mark1(41) -> 68 mark1(42) -> 42 mark1(42) -> 69 mark1(43) -> 43 mark1(43) -> 70 mark1(44) -> 44 mark1(44) -> 71 mark1(45) -> 45 mark1(45) -> 72 mark1(46) -> 46 mark1(46) -> 73 mark1(47) -> 47 mark1(47) -> 74 mark1(48) -> 48 mark1(48) -> 75 mark1(49) -> 49 mark1(49) -> 76 mark1(50) -> 50 mark1(50) -> 77 mark1(51) -> 51 mark1(51) -> 81 mark1(52) -> 52 mark1(52) -> 82 mark1(53) -> 53 mark1(53) -> 83 ok1(54) -> 84 ok1(55) -> 84 ok1(56) -> 84 ok1(57) -> 30 ok1(57) -> 57 ok1(58) -> 31 ok1(58) -> 58 ok1(59) -> 32 ok1(59) -> 59 ok1(60) -> 33 ok1(60) -> 60 ok1(61) -> 34 ok1(61) -> 61 ok1(62) -> 35 ok1(62) -> 62 ok1(63) -> 36 ok1(63) -> 63 ok1(64) -> 37 ok1(64) -> 64 ok1(65) -> 38 ok1(65) -> 65 ok1(66) -> 39 ok1(66) -> 66 ok1(67) -> 40 ok1(67) -> 67 ok1(68) -> 41 ok1(68) -> 68 ok1(69) -> 42 ok1(69) -> 69 ok1(70) -> 43 ok1(70) -> 70 ok1(71) -> 44 ok1(71) -> 71 ok1(72) -> 45 ok1(72) -> 72 ok1(73) -> 46 ok1(73) -> 73 ok1(74) -> 47 ok1(74) -> 74 ok1(75) -> 48 ok1(75) -> 75 ok1(76) -> 49 ok1(76) -> 76 ok1(77) -> 50 ok1(77) -> 77 ok1(78) -> 78 ok1(79) -> 79 ok1(80) -> 80 ok1(81) -> 51 ok1(81) -> 81 ok1(82) -> 52 ok1(82) -> 82 ok1(83) -> 53 ok1(83) -> 83 active2(54) -> 86 top2(86) -> 29 active2(55) -> 86 active2(56) -> 86 ---------------------------------------- (6) BOUNDS(1, n^1) ---------------------------------------- (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 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: FULL ---------------------------------------- (9) 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 [ ]. ---------------------------------------- (10) Complex Obligation (BEST) ---------------------------------------- (11) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 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: FULL ---------------------------------------- (12) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (13) BOUNDS(n^1, INF) ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 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: FULL