/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 754 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 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: a__pairNs -> cons(0, incr(oddNs)) a__oddNs -> a__incr(a__pairNs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__take(0, XS) -> nil a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) a__zip(nil, XS) -> nil a__zip(X, nil) -> nil a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS)) a__tail(cons(X, XS)) -> mark(XS) a__repItems(nil) -> nil a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS))) mark(pairNs) -> a__pairNs mark(incr(X)) -> a__incr(mark(X)) mark(oddNs) -> a__oddNs mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) a__pairNs -> pairNs a__incr(X) -> incr(X) a__oddNs -> oddNs a__take(X1, X2) -> take(X1, X2) a__zip(X1, X2) -> zip(X1, X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(incr(x_1)) -> incr(encArg(x_1)) encArg(oddNs) -> oddNs encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2)) encArg(repItems(x_1)) -> repItems(encArg(x_1)) encArg(pairNs) -> pairNs encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a__pairNs) -> a__pairNs encArg(cons_a__oddNs) -> a__oddNs encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_a__zip(x_1, x_2)) -> a__zip(encArg(x_1), encArg(x_2)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__repItems(x_1)) -> a__repItems(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__pairNs -> a__pairNs encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_incr(x_1) -> incr(encArg(x_1)) encode_oddNs -> oddNs encode_a__oddNs -> a__oddNs encode_a__incr(x_1) -> a__incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_a__zip(x_1, x_2) -> a__zip(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__repItems(x_1) -> a__repItems(encArg(x_1)) encode_repItems(x_1) -> repItems(encArg(x_1)) encode_pairNs -> pairNs encode_tail(x_1) -> tail(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: a__pairNs -> cons(0, incr(oddNs)) a__oddNs -> a__incr(a__pairNs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__take(0, XS) -> nil a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) a__zip(nil, XS) -> nil a__zip(X, nil) -> nil a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS)) a__tail(cons(X, XS)) -> mark(XS) a__repItems(nil) -> nil a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS))) mark(pairNs) -> a__pairNs mark(incr(X)) -> a__incr(mark(X)) mark(oddNs) -> a__oddNs mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) a__pairNs -> pairNs a__incr(X) -> incr(X) a__oddNs -> oddNs a__take(X1, X2) -> take(X1, X2) a__zip(X1, X2) -> zip(X1, X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(incr(x_1)) -> incr(encArg(x_1)) encArg(oddNs) -> oddNs encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2)) encArg(repItems(x_1)) -> repItems(encArg(x_1)) encArg(pairNs) -> pairNs encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a__pairNs) -> a__pairNs encArg(cons_a__oddNs) -> a__oddNs encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_a__zip(x_1, x_2)) -> a__zip(encArg(x_1), encArg(x_2)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__repItems(x_1)) -> a__repItems(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__pairNs -> a__pairNs encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_incr(x_1) -> incr(encArg(x_1)) encode_oddNs -> oddNs encode_a__oddNs -> a__oddNs encode_a__incr(x_1) -> a__incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_a__zip(x_1, x_2) -> a__zip(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__repItems(x_1) -> a__repItems(encArg(x_1)) encode_repItems(x_1) -> repItems(encArg(x_1)) encode_pairNs -> pairNs encode_tail(x_1) -> tail(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: a__pairNs -> cons(0, incr(oddNs)) a__oddNs -> a__incr(a__pairNs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__take(0, XS) -> nil a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) a__zip(nil, XS) -> nil a__zip(X, nil) -> nil a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS)) a__tail(cons(X, XS)) -> mark(XS) a__repItems(nil) -> nil a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS))) mark(pairNs) -> a__pairNs mark(incr(X)) -> a__incr(mark(X)) mark(oddNs) -> a__oddNs mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) a__pairNs -> pairNs a__incr(X) -> incr(X) a__oddNs -> oddNs a__take(X1, X2) -> take(X1, X2) a__zip(X1, X2) -> zip(X1, X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(incr(x_1)) -> incr(encArg(x_1)) encArg(oddNs) -> oddNs encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2)) encArg(repItems(x_1)) -> repItems(encArg(x_1)) encArg(pairNs) -> pairNs encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a__pairNs) -> a__pairNs encArg(cons_a__oddNs) -> a__oddNs encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_a__zip(x_1, x_2)) -> a__zip(encArg(x_1), encArg(x_2)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__repItems(x_1)) -> a__repItems(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__pairNs -> a__pairNs encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_incr(x_1) -> incr(encArg(x_1)) encode_oddNs -> oddNs encode_a__oddNs -> a__oddNs encode_a__incr(x_1) -> a__incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_a__zip(x_1, x_2) -> a__zip(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__repItems(x_1) -> a__repItems(encArg(x_1)) encode_repItems(x_1) -> repItems(encArg(x_1)) encode_pairNs -> pairNs encode_tail(x_1) -> tail(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: a__pairNs -> cons(0, incr(oddNs)) a__oddNs -> a__incr(a__pairNs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__take(0, XS) -> nil a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) a__zip(nil, XS) -> nil a__zip(X, nil) -> nil a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS)) a__tail(cons(X, XS)) -> mark(XS) a__repItems(nil) -> nil a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS))) mark(pairNs) -> a__pairNs mark(incr(X)) -> a__incr(mark(X)) mark(oddNs) -> a__oddNs mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) a__pairNs -> pairNs a__incr(X) -> incr(X) a__oddNs -> oddNs a__take(X1, X2) -> take(X1, X2) a__zip(X1, X2) -> zip(X1, X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(incr(x_1)) -> incr(encArg(x_1)) encArg(oddNs) -> oddNs encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2)) encArg(repItems(x_1)) -> repItems(encArg(x_1)) encArg(pairNs) -> pairNs encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a__pairNs) -> a__pairNs encArg(cons_a__oddNs) -> a__oddNs encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_a__zip(x_1, x_2)) -> a__zip(encArg(x_1), encArg(x_2)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__repItems(x_1)) -> a__repItems(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__pairNs -> a__pairNs encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_incr(x_1) -> incr(encArg(x_1)) encode_oddNs -> oddNs encode_a__oddNs -> a__oddNs encode_a__incr(x_1) -> a__incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_a__zip(x_1, x_2) -> a__zip(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__repItems(x_1) -> a__repItems(encArg(x_1)) encode_repItems(x_1) -> repItems(encArg(x_1)) encode_pairNs -> pairNs encode_tail(x_1) -> tail(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 mark(repItems(X)) ->^+ a__repItems(mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / repItems(X)]. 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: a__pairNs -> cons(0, incr(oddNs)) a__oddNs -> a__incr(a__pairNs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__take(0, XS) -> nil a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) a__zip(nil, XS) -> nil a__zip(X, nil) -> nil a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS)) a__tail(cons(X, XS)) -> mark(XS) a__repItems(nil) -> nil a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS))) mark(pairNs) -> a__pairNs mark(incr(X)) -> a__incr(mark(X)) mark(oddNs) -> a__oddNs mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) a__pairNs -> pairNs a__incr(X) -> incr(X) a__oddNs -> oddNs a__take(X1, X2) -> take(X1, X2) a__zip(X1, X2) -> zip(X1, X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(incr(x_1)) -> incr(encArg(x_1)) encArg(oddNs) -> oddNs encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2)) encArg(repItems(x_1)) -> repItems(encArg(x_1)) encArg(pairNs) -> pairNs encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a__pairNs) -> a__pairNs encArg(cons_a__oddNs) -> a__oddNs encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_a__zip(x_1, x_2)) -> a__zip(encArg(x_1), encArg(x_2)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__repItems(x_1)) -> a__repItems(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__pairNs -> a__pairNs encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_incr(x_1) -> incr(encArg(x_1)) encode_oddNs -> oddNs encode_a__oddNs -> a__oddNs encode_a__incr(x_1) -> a__incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_a__zip(x_1, x_2) -> a__zip(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__repItems(x_1) -> a__repItems(encArg(x_1)) encode_repItems(x_1) -> repItems(encArg(x_1)) encode_pairNs -> pairNs encode_tail(x_1) -> tail(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: a__pairNs -> cons(0, incr(oddNs)) a__oddNs -> a__incr(a__pairNs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__take(0, XS) -> nil a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) a__zip(nil, XS) -> nil a__zip(X, nil) -> nil a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS)) a__tail(cons(X, XS)) -> mark(XS) a__repItems(nil) -> nil a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS))) mark(pairNs) -> a__pairNs mark(incr(X)) -> a__incr(mark(X)) mark(oddNs) -> a__oddNs mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) a__pairNs -> pairNs a__incr(X) -> incr(X) a__oddNs -> oddNs a__take(X1, X2) -> take(X1, X2) a__zip(X1, X2) -> zip(X1, X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(incr(x_1)) -> incr(encArg(x_1)) encArg(oddNs) -> oddNs encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2)) encArg(repItems(x_1)) -> repItems(encArg(x_1)) encArg(pairNs) -> pairNs encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a__pairNs) -> a__pairNs encArg(cons_a__oddNs) -> a__oddNs encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_a__zip(x_1, x_2)) -> a__zip(encArg(x_1), encArg(x_2)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__repItems(x_1)) -> a__repItems(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__pairNs -> a__pairNs encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_incr(x_1) -> incr(encArg(x_1)) encode_oddNs -> oddNs encode_a__oddNs -> a__oddNs encode_a__incr(x_1) -> a__incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_a__zip(x_1, x_2) -> a__zip(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__repItems(x_1) -> a__repItems(encArg(x_1)) encode_repItems(x_1) -> repItems(encArg(x_1)) encode_pairNs -> pairNs encode_tail(x_1) -> tail(encArg(x_1)) Rewrite Strategy: INNERMOST