/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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, 179 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: pairNs -> cons(0, n__incr(n__oddNs)) oddNs -> incr(pairNs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) zip(nil, XS) -> nil zip(X, nil) -> nil zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), n__zip(activate(XS), activate(YS))) tail(cons(X, XS)) -> activate(XS) repItems(nil) -> nil repItems(cons(X, XS)) -> cons(X, n__cons(X, n__repItems(activate(XS)))) incr(X) -> n__incr(X) oddNs -> n__oddNs take(X1, X2) -> n__take(X1, X2) zip(X1, X2) -> n__zip(X1, X2) cons(X1, X2) -> n__cons(X1, X2) repItems(X) -> n__repItems(X) activate(n__incr(X)) -> incr(activate(X)) activate(n__oddNs) -> oddNs activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__zip(X1, X2)) -> zip(activate(X1), activate(X2)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__repItems(X)) -> repItems(activate(X)) activate(X) -> X 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: pairNs -> cons(0, n__incr(n__oddNs)) oddNs -> incr(pairNs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) zip(nil, XS) -> nil zip(X, nil) -> nil zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), n__zip(activate(XS), activate(YS))) tail(cons(X, XS)) -> activate(XS) repItems(nil) -> nil repItems(cons(X, XS)) -> cons(X, n__cons(X, n__repItems(activate(XS)))) incr(X) -> n__incr(X) oddNs -> n__oddNs take(X1, X2) -> n__take(X1, X2) zip(X1, X2) -> n__zip(X1, X2) cons(X1, X2) -> n__cons(X1, X2) repItems(X) -> n__repItems(X) activate(n__incr(X)) -> incr(activate(X)) activate(n__oddNs) -> oddNs activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__zip(X1, X2)) -> zip(activate(X1), activate(X2)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__repItems(X)) -> repItems(activate(X)) activate(X) -> X 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 activate(n__cons(X1, X2)) ->^+ cons(activate(X1), X2) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / n__cons(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: pairNs -> cons(0, n__incr(n__oddNs)) oddNs -> incr(pairNs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) zip(nil, XS) -> nil zip(X, nil) -> nil zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), n__zip(activate(XS), activate(YS))) tail(cons(X, XS)) -> activate(XS) repItems(nil) -> nil repItems(cons(X, XS)) -> cons(X, n__cons(X, n__repItems(activate(XS)))) incr(X) -> n__incr(X) oddNs -> n__oddNs take(X1, X2) -> n__take(X1, X2) zip(X1, X2) -> n__zip(X1, X2) cons(X1, X2) -> n__cons(X1, X2) repItems(X) -> n__repItems(X) activate(n__incr(X)) -> incr(activate(X)) activate(n__oddNs) -> oddNs activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__zip(X1, X2)) -> zip(activate(X1), activate(X2)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__repItems(X)) -> repItems(activate(X)) activate(X) -> X 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: pairNs -> cons(0, n__incr(n__oddNs)) oddNs -> incr(pairNs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) zip(nil, XS) -> nil zip(X, nil) -> nil zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), n__zip(activate(XS), activate(YS))) tail(cons(X, XS)) -> activate(XS) repItems(nil) -> nil repItems(cons(X, XS)) -> cons(X, n__cons(X, n__repItems(activate(XS)))) incr(X) -> n__incr(X) oddNs -> n__oddNs take(X1, X2) -> n__take(X1, X2) zip(X1, X2) -> n__zip(X1, X2) cons(X1, X2) -> n__cons(X1, X2) repItems(X) -> n__repItems(X) activate(n__incr(X)) -> incr(activate(X)) activate(n__oddNs) -> oddNs activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__zip(X1, X2)) -> zip(activate(X1), activate(X2)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__repItems(X)) -> repItems(activate(X)) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence activate(n__repItems(n__cons(X11_0, X22_0))) ->^+ cons(activate(X11_0), n__cons(activate(X11_0), n__repItems(activate(X22_0)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X11_0 / n__repItems(n__cons(X11_0, X22_0))]. The result substitution is [ ]. The rewrite sequence activate(n__repItems(n__cons(X11_0, X22_0))) ->^+ cons(activate(X11_0), n__cons(activate(X11_0), n__repItems(activate(X22_0)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0]. The pumping substitution is [X11_0 / n__repItems(n__cons(X11_0, X22_0))]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)