/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 502 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 7 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__from(X) -> cons(mark(X), from(s(X))) a__sel(0, cons(X, XS)) -> mark(X) a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) a__minus(X, 0) -> 0 a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y)) a__quot(0, s(Y)) -> 0 a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y)))) a__zWquot(XS, nil) -> nil a__zWquot(nil, XS) -> nil a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS)) mark(from(X)) -> a__from(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2)) mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2)) mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__from(X) -> from(X) a__sel(X1, X2) -> sel(X1, X2) a__minus(X1, X2) -> minus(X1, X2) a__quot(X1, X2) -> quot(X1, X2) a__zWquot(X1, X2) -> zWquot(X1, X2) 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(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__minus(x_1, x_2)) -> a__minus(encArg(x_1), encArg(x_2)) encArg(cons_a__quot(x_1, x_2)) -> a__quot(encArg(x_1), encArg(x_2)) encArg(cons_a__zWquot(x_1, x_2)) -> a__zWquot(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_a__minus(x_1, x_2) -> a__minus(encArg(x_1), encArg(x_2)) encode_a__quot(x_1, x_2) -> a__quot(encArg(x_1), encArg(x_2)) encode_a__zWquot(x_1, x_2) -> a__zWquot(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) ---------------------------------------- (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__from(X) -> cons(mark(X), from(s(X))) a__sel(0, cons(X, XS)) -> mark(X) a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) a__minus(X, 0) -> 0 a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y)) a__quot(0, s(Y)) -> 0 a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y)))) a__zWquot(XS, nil) -> nil a__zWquot(nil, XS) -> nil a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS)) mark(from(X)) -> a__from(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2)) mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2)) mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__from(X) -> from(X) a__sel(X1, X2) -> sel(X1, X2) a__minus(X1, X2) -> minus(X1, X2) a__quot(X1, X2) -> quot(X1, X2) a__zWquot(X1, X2) -> zWquot(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__minus(x_1, x_2)) -> a__minus(encArg(x_1), encArg(x_2)) encArg(cons_a__quot(x_1, x_2)) -> a__quot(encArg(x_1), encArg(x_2)) encArg(cons_a__zWquot(x_1, x_2)) -> a__zWquot(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_a__minus(x_1, x_2) -> a__minus(encArg(x_1), encArg(x_2)) encode_a__quot(x_1, x_2) -> a__quot(encArg(x_1), encArg(x_2)) encode_a__zWquot(x_1, x_2) -> a__zWquot(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(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(n^1, INF). The TRS R consists of the following rules: a__from(X) -> cons(mark(X), from(s(X))) a__sel(0, cons(X, XS)) -> mark(X) a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) a__minus(X, 0) -> 0 a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y)) a__quot(0, s(Y)) -> 0 a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y)))) a__zWquot(XS, nil) -> nil a__zWquot(nil, XS) -> nil a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS)) mark(from(X)) -> a__from(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2)) mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2)) mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__from(X) -> from(X) a__sel(X1, X2) -> sel(X1, X2) a__minus(X1, X2) -> minus(X1, X2) a__quot(X1, X2) -> quot(X1, X2) a__zWquot(X1, X2) -> zWquot(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__minus(x_1, x_2)) -> a__minus(encArg(x_1), encArg(x_2)) encArg(cons_a__quot(x_1, x_2)) -> a__quot(encArg(x_1), encArg(x_2)) encArg(cons_a__zWquot(x_1, x_2)) -> a__zWquot(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_a__minus(x_1, x_2) -> a__minus(encArg(x_1), encArg(x_2)) encode_a__quot(x_1, x_2) -> a__quot(encArg(x_1), encArg(x_2)) encode_a__zWquot(x_1, x_2) -> a__zWquot(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(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(n^1, INF). The TRS R consists of the following rules: a__from(X) -> cons(mark(X), from(s(X))) a__sel(0, cons(X, XS)) -> mark(X) a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) a__minus(X, 0) -> 0 a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y)) a__quot(0, s(Y)) -> 0 a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y)))) a__zWquot(XS, nil) -> nil a__zWquot(nil, XS) -> nil a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS)) mark(from(X)) -> a__from(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2)) mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2)) mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__from(X) -> from(X) a__sel(X1, X2) -> sel(X1, X2) a__minus(X1, X2) -> minus(X1, X2) a__quot(X1, X2) -> quot(X1, X2) a__zWquot(X1, X2) -> zWquot(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__minus(x_1, x_2)) -> a__minus(encArg(x_1), encArg(x_2)) encArg(cons_a__quot(x_1, x_2)) -> a__quot(encArg(x_1), encArg(x_2)) encArg(cons_a__zWquot(x_1, x_2)) -> a__zWquot(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_a__minus(x_1, x_2) -> a__minus(encArg(x_1), encArg(x_2)) encode_a__quot(x_1, x_2) -> a__quot(encArg(x_1), encArg(x_2)) encode_a__zWquot(x_1, x_2) -> a__zWquot(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(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 mark(from(X)) ->^+ a__from(mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / from(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__from(X) -> cons(mark(X), from(s(X))) a__sel(0, cons(X, XS)) -> mark(X) a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) a__minus(X, 0) -> 0 a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y)) a__quot(0, s(Y)) -> 0 a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y)))) a__zWquot(XS, nil) -> nil a__zWquot(nil, XS) -> nil a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS)) mark(from(X)) -> a__from(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2)) mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2)) mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__from(X) -> from(X) a__sel(X1, X2) -> sel(X1, X2) a__minus(X1, X2) -> minus(X1, X2) a__quot(X1, X2) -> quot(X1, X2) a__zWquot(X1, X2) -> zWquot(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__minus(x_1, x_2)) -> a__minus(encArg(x_1), encArg(x_2)) encArg(cons_a__quot(x_1, x_2)) -> a__quot(encArg(x_1), encArg(x_2)) encArg(cons_a__zWquot(x_1, x_2)) -> a__zWquot(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_a__minus(x_1, x_2) -> a__minus(encArg(x_1), encArg(x_2)) encode_a__quot(x_1, x_2) -> a__quot(encArg(x_1), encArg(x_2)) encode_a__zWquot(x_1, x_2) -> a__zWquot(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(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(n^1, INF). The TRS R consists of the following rules: a__from(X) -> cons(mark(X), from(s(X))) a__sel(0, cons(X, XS)) -> mark(X) a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) a__minus(X, 0) -> 0 a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y)) a__quot(0, s(Y)) -> 0 a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y)))) a__zWquot(XS, nil) -> nil a__zWquot(nil, XS) -> nil a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS)) mark(from(X)) -> a__from(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2)) mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2)) mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__from(X) -> from(X) a__sel(X1, X2) -> sel(X1, X2) a__minus(X1, X2) -> minus(X1, X2) a__quot(X1, X2) -> quot(X1, X2) a__zWquot(X1, X2) -> zWquot(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__minus(x_1, x_2)) -> a__minus(encArg(x_1), encArg(x_2)) encArg(cons_a__quot(x_1, x_2)) -> a__quot(encArg(x_1), encArg(x_2)) encArg(cons_a__zWquot(x_1, x_2)) -> a__zWquot(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_a__minus(x_1, x_2) -> a__minus(encArg(x_1), encArg(x_2)) encode_a__quot(x_1, x_2) -> a__quot(encArg(x_1), encArg(x_2)) encode_a__zWquot(x_1, x_2) -> a__zWquot(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST