/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), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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 ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt) -> tt a__U21(tt) -> tt a__U31(tt) -> tt a__U41(tt, V2) -> a__U42(a__isNatIList(V2)) a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatList(V2)) a__U52(tt) -> tt a__U61(tt, L, N) -> a__U62(a__isNat(N), L) a__U62(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) mark(U62(X1, X2)) -> a__U62(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) 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(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1, X2) -> U41(X1, X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1, X2, X3) -> U61(X1, X2, X3) a__U62(X1, X2) -> U62(X1, X2) a__isNat(X) -> isNat(X) a__length(X) -> length(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(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt) -> tt a__U21(tt) -> tt a__U31(tt) -> tt a__U41(tt, V2) -> a__U42(a__isNatIList(V2)) a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatList(V2)) a__U52(tt) -> tt a__U61(tt, L, N) -> a__U62(a__isNat(N), L) a__U62(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) mark(U62(X1, X2)) -> a__U62(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) 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(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1, X2) -> U41(X1, X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1, X2, X3) -> U61(X1, X2, X3) a__U62(X1, X2) -> U62(X1, X2) a__isNat(X) -> isNat(X) a__length(X) -> length(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 mark(U61(X1, X2, X3)) ->^+ a__U61(mark(X1), X2, X3) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / U61(X1, X2, X3)]. 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(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt) -> tt a__U21(tt) -> tt a__U31(tt) -> tt a__U41(tt, V2) -> a__U42(a__isNatIList(V2)) a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatList(V2)) a__U52(tt) -> tt a__U61(tt, L, N) -> a__U62(a__isNat(N), L) a__U62(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) mark(U62(X1, X2)) -> a__U62(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) 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(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1, X2) -> U41(X1, X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1, X2, X3) -> U61(X1, X2, X3) a__U62(X1, X2) -> U62(X1, X2) a__isNat(X) -> isNat(X) a__length(X) -> length(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(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt) -> tt a__U21(tt) -> tt a__U31(tt) -> tt a__U41(tt, V2) -> a__U42(a__isNatIList(V2)) a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatList(V2)) a__U52(tt) -> tt a__U61(tt, L, N) -> a__U62(a__isNat(N), L) a__U62(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) mark(U62(X1, X2)) -> a__U62(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) 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(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1, X2) -> U41(X1, X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1, X2, X3) -> U61(X1, X2, X3) a__U62(X1, X2) -> U62(X1, X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) S is empty. Rewrite Strategy: FULL