/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1558 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 3 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection (13) InfiniteLowerBoundProof [FINISHED, 3162 ms] (14) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) sel(0, cons(X, Z)) -> X first(0, Z) -> nil first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) from(X) -> cons(X, n__from(s(X))) sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z)) sel1(0, cons(X, Z)) -> quote(X) first1(0, Z) -> nil1 first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z))) quote(n__0) -> 01 quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z))) quote1(n__nil) -> nil1 quote(n__s(X)) -> s1(quote(activate(X))) quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z)) quote1(n__first(X, Z)) -> first1(activate(X), activate(Z)) unquote(01) -> 0 unquote(s1(X)) -> s(unquote(X)) unquote1(nil1) -> nil unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z)) fcons(X, Z) -> cons(X, Z) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) activate(n__first(X1, X2)) -> first(X1, X2) activate(n__from(X)) -> from(X) activate(n__0) -> 0 activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) 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__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(nil1) -> nil1 encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(n__0) -> n__0 encArg(01) -> 01 encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(n__sel(x_1, x_2)) -> n__sel(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_first1(x_1, x_2)) -> first1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_quote1(x_1)) -> quote1(encArg(x_1)) encArg(cons_unquote(x_1)) -> unquote(encArg(x_1)) encArg(cons_unquote1(x_1)) -> unquote1(encArg(x_1)) encArg(cons_fcons(x_1, x_2)) -> fcons(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_0 -> 0 encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_first1(x_1, x_2) -> first1(encArg(x_1), encArg(x_2)) encode_nil1 -> nil1 encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_01 -> 01 encode_quote1(x_1) -> quote1(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s1(x_1) -> s1(encArg(x_1)) encode_n__sel(x_1, x_2) -> n__sel(encArg(x_1), encArg(x_2)) encode_unquote(x_1) -> unquote(encArg(x_1)) encode_unquote1(x_1) -> unquote1(encArg(x_1)) encode_fcons(x_1, x_2) -> fcons(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) sel(0, cons(X, Z)) -> X first(0, Z) -> nil first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) from(X) -> cons(X, n__from(s(X))) sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z)) sel1(0, cons(X, Z)) -> quote(X) first1(0, Z) -> nil1 first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z))) quote(n__0) -> 01 quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z))) quote1(n__nil) -> nil1 quote(n__s(X)) -> s1(quote(activate(X))) quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z)) quote1(n__first(X, Z)) -> first1(activate(X), activate(Z)) unquote(01) -> 0 unquote(s1(X)) -> s(unquote(X)) unquote1(nil1) -> nil unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z)) fcons(X, Z) -> cons(X, Z) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) activate(n__first(X1, X2)) -> first(X1, X2) activate(n__from(X)) -> from(X) activate(n__0) -> 0 activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(nil1) -> nil1 encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(n__0) -> n__0 encArg(01) -> 01 encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(n__sel(x_1, x_2)) -> n__sel(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_first1(x_1, x_2)) -> first1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_quote1(x_1)) -> quote1(encArg(x_1)) encArg(cons_unquote(x_1)) -> unquote(encArg(x_1)) encArg(cons_unquote1(x_1)) -> unquote1(encArg(x_1)) encArg(cons_fcons(x_1, x_2)) -> fcons(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_0 -> 0 encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_first1(x_1, x_2) -> first1(encArg(x_1), encArg(x_2)) encode_nil1 -> nil1 encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_01 -> 01 encode_quote1(x_1) -> quote1(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s1(x_1) -> s1(encArg(x_1)) encode_n__sel(x_1, x_2) -> n__sel(encArg(x_1), encArg(x_2)) encode_unquote(x_1) -> unquote(encArg(x_1)) encode_unquote1(x_1) -> unquote1(encArg(x_1)) encode_fcons(x_1, x_2) -> fcons(encArg(x_1), encArg(x_2)) 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(INF, INF). The TRS R consists of the following rules: sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) sel(0, cons(X, Z)) -> X first(0, Z) -> nil first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) from(X) -> cons(X, n__from(s(X))) sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z)) sel1(0, cons(X, Z)) -> quote(X) first1(0, Z) -> nil1 first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z))) quote(n__0) -> 01 quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z))) quote1(n__nil) -> nil1 quote(n__s(X)) -> s1(quote(activate(X))) quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z)) quote1(n__first(X, Z)) -> first1(activate(X), activate(Z)) unquote(01) -> 0 unquote(s1(X)) -> s(unquote(X)) unquote1(nil1) -> nil unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z)) fcons(X, Z) -> cons(X, Z) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) activate(n__first(X1, X2)) -> first(X1, X2) activate(n__from(X)) -> from(X) activate(n__0) -> 0 activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(nil1) -> nil1 encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(n__0) -> n__0 encArg(01) -> 01 encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(n__sel(x_1, x_2)) -> n__sel(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_first1(x_1, x_2)) -> first1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_quote1(x_1)) -> quote1(encArg(x_1)) encArg(cons_unquote(x_1)) -> unquote(encArg(x_1)) encArg(cons_unquote1(x_1)) -> unquote1(encArg(x_1)) encArg(cons_fcons(x_1, x_2)) -> fcons(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_0 -> 0 encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_first1(x_1, x_2) -> first1(encArg(x_1), encArg(x_2)) encode_nil1 -> nil1 encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_01 -> 01 encode_quote1(x_1) -> quote1(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s1(x_1) -> s1(encArg(x_1)) encode_n__sel(x_1, x_2) -> n__sel(encArg(x_1), encArg(x_2)) encode_unquote(x_1) -> unquote(encArg(x_1)) encode_unquote1(x_1) -> unquote1(encArg(x_1)) encode_fcons(x_1, x_2) -> fcons(encArg(x_1), encArg(x_2)) 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(INF, INF). The TRS R consists of the following rules: sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) sel(0, cons(X, Z)) -> X first(0, Z) -> nil first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) from(X) -> cons(X, n__from(s(X))) sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z)) sel1(0, cons(X, Z)) -> quote(X) first1(0, Z) -> nil1 first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z))) quote(n__0) -> 01 quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z))) quote1(n__nil) -> nil1 quote(n__s(X)) -> s1(quote(activate(X))) quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z)) quote1(n__first(X, Z)) -> first1(activate(X), activate(Z)) unquote(01) -> 0 unquote(s1(X)) -> s(unquote(X)) unquote1(nil1) -> nil unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z)) fcons(X, Z) -> cons(X, Z) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) activate(n__first(X1, X2)) -> first(X1, X2) activate(n__from(X)) -> from(X) activate(n__0) -> 0 activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(nil1) -> nil1 encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(n__0) -> n__0 encArg(01) -> 01 encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(n__sel(x_1, x_2)) -> n__sel(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_first1(x_1, x_2)) -> first1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_quote1(x_1)) -> quote1(encArg(x_1)) encArg(cons_unquote(x_1)) -> unquote(encArg(x_1)) encArg(cons_unquote1(x_1)) -> unquote1(encArg(x_1)) encArg(cons_fcons(x_1, x_2)) -> fcons(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_0 -> 0 encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_first1(x_1, x_2) -> first1(encArg(x_1), encArg(x_2)) encode_nil1 -> nil1 encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_01 -> 01 encode_quote1(x_1) -> quote1(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s1(x_1) -> s1(encArg(x_1)) encode_n__sel(x_1, x_2) -> n__sel(encArg(x_1), encArg(x_2)) encode_unquote(x_1) -> unquote(encArg(x_1)) encode_unquote1(x_1) -> unquote1(encArg(x_1)) encode_fcons(x_1, x_2) -> fcons(encArg(x_1), encArg(x_2)) 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 unquote1(cons1(X, Z)) ->^+ fcons(unquote(X), unquote1(Z)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [Z / cons1(X, Z)]. 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(INF, INF). The TRS R consists of the following rules: sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) sel(0, cons(X, Z)) -> X first(0, Z) -> nil first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) from(X) -> cons(X, n__from(s(X))) sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z)) sel1(0, cons(X, Z)) -> quote(X) first1(0, Z) -> nil1 first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z))) quote(n__0) -> 01 quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z))) quote1(n__nil) -> nil1 quote(n__s(X)) -> s1(quote(activate(X))) quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z)) quote1(n__first(X, Z)) -> first1(activate(X), activate(Z)) unquote(01) -> 0 unquote(s1(X)) -> s(unquote(X)) unquote1(nil1) -> nil unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z)) fcons(X, Z) -> cons(X, Z) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) activate(n__first(X1, X2)) -> first(X1, X2) activate(n__from(X)) -> from(X) activate(n__0) -> 0 activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(nil1) -> nil1 encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(n__0) -> n__0 encArg(01) -> 01 encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(n__sel(x_1, x_2)) -> n__sel(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_first1(x_1, x_2)) -> first1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_quote1(x_1)) -> quote1(encArg(x_1)) encArg(cons_unquote(x_1)) -> unquote(encArg(x_1)) encArg(cons_unquote1(x_1)) -> unquote1(encArg(x_1)) encArg(cons_fcons(x_1, x_2)) -> fcons(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_0 -> 0 encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_first1(x_1, x_2) -> first1(encArg(x_1), encArg(x_2)) encode_nil1 -> nil1 encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_01 -> 01 encode_quote1(x_1) -> quote1(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s1(x_1) -> s1(encArg(x_1)) encode_n__sel(x_1, x_2) -> n__sel(encArg(x_1), encArg(x_2)) encode_unquote(x_1) -> unquote(encArg(x_1)) encode_unquote1(x_1) -> unquote1(encArg(x_1)) encode_fcons(x_1, x_2) -> fcons(encArg(x_1), encArg(x_2)) 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(INF, INF). The TRS R consists of the following rules: sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) sel(0, cons(X, Z)) -> X first(0, Z) -> nil first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) from(X) -> cons(X, n__from(s(X))) sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z)) sel1(0, cons(X, Z)) -> quote(X) first1(0, Z) -> nil1 first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z))) quote(n__0) -> 01 quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z))) quote1(n__nil) -> nil1 quote(n__s(X)) -> s1(quote(activate(X))) quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z)) quote1(n__first(X, Z)) -> first1(activate(X), activate(Z)) unquote(01) -> 0 unquote(s1(X)) -> s(unquote(X)) unquote1(nil1) -> nil unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z)) fcons(X, Z) -> cons(X, Z) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) activate(n__first(X1, X2)) -> first(X1, X2) activate(n__from(X)) -> from(X) activate(n__0) -> 0 activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(nil1) -> nil1 encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(n__0) -> n__0 encArg(01) -> 01 encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(n__sel(x_1, x_2)) -> n__sel(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_first1(x_1, x_2)) -> first1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_quote1(x_1)) -> quote1(encArg(x_1)) encArg(cons_unquote(x_1)) -> unquote(encArg(x_1)) encArg(cons_unquote1(x_1)) -> unquote1(encArg(x_1)) encArg(cons_fcons(x_1, x_2)) -> fcons(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_0 -> 0 encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_first1(x_1, x_2) -> first1(encArg(x_1), encArg(x_2)) encode_nil1 -> nil1 encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_01 -> 01 encode_quote1(x_1) -> quote1(encArg(x_1)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s1(x_1) -> s1(encArg(x_1)) encode_n__sel(x_1, x_2) -> n__sel(encArg(x_1), encArg(x_2)) encode_unquote(x_1) -> unquote(encArg(x_1)) encode_unquote1(x_1) -> unquote1(encArg(x_1)) encode_fcons(x_1, x_2) -> fcons(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) InfiniteLowerBoundProof (FINISHED) The following loop proves infinite runtime complexity: The rewrite sequence quote1(n__cons(X, n__from(X1_0))) ->^+ cons1(quote(activate(X)), quote1(n__cons(X1_0, n__from(n__s(X1_0))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [ ]. The result substitution is [X / X1_0, X1_0 / n__s(X1_0)]. ---------------------------------------- (14) BOUNDS(INF, INF)