/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1090 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 108 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) 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__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(n__zeros) -> n__zeros encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__isNatIList(x_1)) -> n__isNatIList(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__isNatList(x_1)) -> n__isNatList(encArg(x_1)) encArg(n__isNat(x_1)) -> n__isNat(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__zeros -> n__zeros encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__length(x_1) -> n__length(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__isNatIList(x_1) -> n__isNatIList(encArg(x_1)) encode_n__nil -> n__nil encode_n__isNatList(x_1) -> n__isNatList(encArg(x_1)) encode_nil -> nil encode_n__isNat(x_1) -> n__isNat(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) 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__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__zeros) -> n__zeros encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__isNatIList(x_1)) -> n__isNatIList(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__isNatList(x_1)) -> n__isNatList(encArg(x_1)) encArg(n__isNat(x_1)) -> n__isNat(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__zeros -> n__zeros encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__length(x_1) -> n__length(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__isNatIList(x_1) -> n__isNatIList(encArg(x_1)) encode_n__nil -> n__nil encode_n__isNatList(x_1) -> n__isNatList(encArg(x_1)) encode_nil -> nil encode_n__isNat(x_1) -> n__isNat(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) 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__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__zeros) -> n__zeros encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__isNatIList(x_1)) -> n__isNatIList(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__isNatList(x_1)) -> n__isNatList(encArg(x_1)) encArg(n__isNat(x_1)) -> n__isNat(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__zeros -> n__zeros encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__length(x_1) -> n__length(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__isNatIList(x_1) -> n__isNatIList(encArg(x_1)) encode_n__nil -> n__nil encode_n__isNatList(x_1) -> n__isNatList(encArg(x_1)) encode_nil -> nil encode_n__isNat(x_1) -> n__isNat(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) 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__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__zeros) -> n__zeros encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__isNatIList(x_1)) -> n__isNatIList(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__isNatList(x_1)) -> n__isNatList(encArg(x_1)) encArg(n__isNat(x_1)) -> n__isNat(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__zeros -> n__zeros encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__length(x_1) -> n__length(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__isNatIList(x_1) -> n__isNatIList(encArg(x_1)) encode_n__nil -> n__nil encode_n__isNatList(x_1) -> n__isNatList(encArg(x_1)) encode_nil -> nil encode_n__isNat(x_1) -> n__isNat(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence activate(n__isNatList(n__cons(V11_0, V22_0))) ->^+ and(isNat(activate(V11_0)), n__isNatList(activate(V22_0))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [V11_0 / n__isNatList(n__cons(V11_0, V22_0))]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) 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__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__zeros) -> n__zeros encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__isNatIList(x_1)) -> n__isNatIList(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__isNatList(x_1)) -> n__isNatList(encArg(x_1)) encArg(n__isNat(x_1)) -> n__isNat(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__zeros -> n__zeros encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__length(x_1) -> n__length(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__isNatIList(x_1) -> n__isNatIList(encArg(x_1)) encode_n__nil -> n__nil encode_n__isNatList(x_1) -> n__isNatList(encArg(x_1)) encode_nil -> nil encode_n__isNat(x_1) -> n__isNat(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) 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__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__zeros) -> n__zeros encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__isNatIList(x_1)) -> n__isNatIList(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__isNatList(x_1)) -> n__isNatList(encArg(x_1)) encArg(n__isNat(x_1)) -> n__isNat(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__zeros -> n__zeros encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__length(x_1) -> n__length(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__isNatIList(x_1) -> n__isNatIList(encArg(x_1)) encode_n__nil -> n__nil encode_n__isNatList(x_1) -> n__isNatList(encArg(x_1)) encode_nil -> nil encode_n__isNat(x_1) -> n__isNat(encArg(x_1)) Rewrite Strategy: INNERMOST