/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 241 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: zeros -> cons(0, n__zeros) U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U12(tt, V1) -> U13(isNatList(activate(V1))) U13(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt) -> tt U81(tt, V1, V2) -> U82(isNatKind(activate(V1)), activate(V1), activate(V2)) U82(tt, V1, V2) -> U83(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U83(tt, V1, V2) -> U84(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U84(tt, V1, V2) -> U85(isNat(activate(V1)), activate(V2)) U85(tt, V2) -> U86(isNatList(activate(V2))) U86(tt) -> tt U91(tt, L, N) -> U92(isNatIListKind(activate(L)), activate(L), activate(N)) U92(tt, L, N) -> U93(isNat(activate(N)), activate(L), activate(N)) U93(tt, L, N) -> U94(isNatKind(activate(N)), activate(L)) U94(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U61(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U71(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U81(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U91(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil 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(n^1, INF). The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U12(tt, V1) -> U13(isNatList(activate(V1))) U13(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt) -> tt U81(tt, V1, V2) -> U82(isNatKind(activate(V1)), activate(V1), activate(V2)) U82(tt, V1, V2) -> U83(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U83(tt, V1, V2) -> U84(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U84(tt, V1, V2) -> U85(isNat(activate(V1)), activate(V2)) U85(tt, V2) -> U86(isNatList(activate(V2))) U86(tt) -> tt U91(tt, L, N) -> U92(isNatIListKind(activate(L)), activate(L), activate(N)) U92(tt, L, N) -> U93(isNat(activate(N)), activate(L), activate(N)) U93(tt, L, N) -> U94(isNatKind(activate(N)), activate(L)) U94(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U61(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U71(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U81(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U91(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil 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 isNatKind(n__s(V1)) ->^+ U71(isNatKind(V1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [V1 / n__s(V1)]. 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: zeros -> cons(0, n__zeros) U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U12(tt, V1) -> U13(isNatList(activate(V1))) U13(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt) -> tt U81(tt, V1, V2) -> U82(isNatKind(activate(V1)), activate(V1), activate(V2)) U82(tt, V1, V2) -> U83(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U83(tt, V1, V2) -> U84(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U84(tt, V1, V2) -> U85(isNat(activate(V1)), activate(V2)) U85(tt, V2) -> U86(isNatList(activate(V2))) U86(tt) -> tt U91(tt, L, N) -> U92(isNatIListKind(activate(L)), activate(L), activate(N)) U92(tt, L, N) -> U93(isNat(activate(N)), activate(L), activate(N)) U93(tt, L, N) -> U94(isNatKind(activate(N)), activate(L)) U94(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U61(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U71(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U81(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U91(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil 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(n^1, INF). The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U12(tt, V1) -> U13(isNatList(activate(V1))) U13(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt) -> tt U71(tt) -> tt U81(tt, V1, V2) -> U82(isNatKind(activate(V1)), activate(V1), activate(V2)) U82(tt, V1, V2) -> U83(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U83(tt, V1, V2) -> U84(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U84(tt, V1, V2) -> U85(isNat(activate(V1)), activate(V2)) U85(tt, V2) -> U86(isNatList(activate(V2))) U86(tt) -> tt U91(tt, L, N) -> U92(isNatIListKind(activate(L)), activate(L), activate(N)) U92(tt, L, N) -> U93(isNat(activate(N)), activate(L), activate(N)) U93(tt, L, N) -> U94(isNatKind(activate(N)), activate(L)) U94(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U61(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U71(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U81(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U91(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X S is empty. Rewrite Strategy: FULL