/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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, 6284 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__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(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__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(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 a__isNat(s(N)) ->^+ a__isNat(N) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [N / s(N)]. 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__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(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__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(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(take(s(X1_0), cons(X11_1, X22_1))) ->^+ a__uTake2(a__and(a__isNat(mark(X1_0)), a__and(a__isNat(mark(X11_1)), a__isNatIList(X22_1))), mark(X1_0), mark(X11_1), X22_1) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0]. The pumping substitution is [X1_0 / take(s(X1_0), cons(X11_1, X22_1))]. The result substitution is [ ]. The rewrite sequence mark(take(s(X1_0), cons(X11_1, X22_1))) ->^+ a__uTake2(a__and(a__isNat(mark(X1_0)), a__and(a__isNat(mark(X11_1)), a__isNatIList(X22_1))), mark(X1_0), mark(X11_1), X22_1) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [X1_0 / take(s(X1_0), cons(X11_1, X22_1))]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)