/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection (9) DecreasingLoopProof [FINISHED, 85 ms] (10) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__and(tt, X) -> mark(X) a__fst(pair(X, Y)) -> mark(X) a__head(cons(N, XS)) -> mark(N) a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__snd(pair(X, Y)) -> mark(Y) a__splitAt(0, XS) -> pair(nil, mark(XS)) a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) a__tail(cons(N, XS)) -> mark(XS) a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(snd(X)) -> a__snd(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(fst(X)) -> a__fst(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(tt) -> tt mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) a__U12(X1, X2) -> U12(X1, X2) a__splitAt(X1, X2) -> splitAt(X1, X2) a__afterNth(X1, X2) -> afterNth(X1, X2) a__snd(X) -> snd(X) a__and(X1, X2) -> and(X1, X2) a__fst(X) -> fst(X) a__head(X) -> head(X) a__natsFrom(X) -> natsFrom(X) a__sel(X1, X2) -> sel(X1, X2) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__and(tt, X) -> mark(X) a__fst(pair(X, Y)) -> mark(X) a__head(cons(N, XS)) -> mark(N) a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__snd(pair(X, Y)) -> mark(Y) a__splitAt(0, XS) -> pair(nil, mark(XS)) a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) a__tail(cons(N, XS)) -> mark(XS) a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(snd(X)) -> a__snd(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(fst(X)) -> a__fst(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(tt) -> tt mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) a__U12(X1, X2) -> U12(X1, X2) a__splitAt(X1, X2) -> splitAt(X1, X2) a__afterNth(X1, X2) -> afterNth(X1, X2) a__snd(X) -> snd(X) a__and(X1, X2) -> and(X1, X2) a__fst(X) -> fst(X) a__head(X) -> head(X) a__natsFrom(X) -> natsFrom(X) a__sel(X1, X2) -> sel(X1, X2) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence mark(afterNth(X1, X2)) ->^+ a__afterNth(mark(X1), mark(X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / afterNth(X1, X2)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) 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(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__and(tt, X) -> mark(X) a__fst(pair(X, Y)) -> mark(X) a__head(cons(N, XS)) -> mark(N) a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__snd(pair(X, Y)) -> mark(Y) a__splitAt(0, XS) -> pair(nil, mark(XS)) a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) a__tail(cons(N, XS)) -> mark(XS) a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(snd(X)) -> a__snd(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(fst(X)) -> a__fst(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(tt) -> tt mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) a__U12(X1, X2) -> U12(X1, X2) a__splitAt(X1, X2) -> splitAt(X1, X2) a__afterNth(X1, X2) -> afterNth(X1, X2) a__snd(X) -> snd(X) a__and(X1, X2) -> and(X1, X2) a__fst(X) -> fst(X) a__head(X) -> head(X) a__natsFrom(X) -> natsFrom(X) a__sel(X1, X2) -> sel(X1, X2) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__and(tt, X) -> mark(X) a__fst(pair(X, Y)) -> mark(X) a__head(cons(N, XS)) -> mark(N) a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__snd(pair(X, Y)) -> mark(Y) a__splitAt(0, XS) -> pair(nil, mark(XS)) a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) a__tail(cons(N, XS)) -> mark(XS) a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(snd(X)) -> a__snd(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(fst(X)) -> a__fst(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(tt) -> tt mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) a__U12(X1, X2) -> U12(X1, X2) a__splitAt(X1, X2) -> splitAt(X1, X2) a__afterNth(X1, X2) -> afterNth(X1, X2) a__snd(X) -> snd(X) a__and(X1, X2) -> and(X1, X2) a__fst(X) -> fst(X) a__head(X) -> head(X) a__natsFrom(X) -> natsFrom(X) a__sel(X1, X2) -> sel(X1, X2) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence mark(natsFrom(X)) ->^+ cons(mark(mark(X)), natsFrom(s(mark(X)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [X / natsFrom(X)]. The result substitution is [ ]. The rewrite sequence mark(natsFrom(X)) ->^+ cons(mark(mark(X)), natsFrom(s(mark(X)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0]. The pumping substitution is [X / natsFrom(X)]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)