/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).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) InfiniteLowerBoundProof [FINISHED, 47.1 s] (10) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 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(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 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(U21(X1, X2, X3, X4)) ->^+ a__U21(mark(X1), X2, X3, X4) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / U21(X1, X2, X3, X4)]. 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(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 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(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) InfiniteLowerBoundProof (FINISHED) The following loop proves infinite runtime complexity: The rewrite sequence a__U11(tt, zeros) ->^+ s(a__U11(tt, zeros)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [ ]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(INF, INF)