/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 608 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: plus(x, y) -> ifPlus(isZero(x), x, inc(y)) ifPlus(true, x, y) -> p(y) ifPlus(false, x, y) -> plus(p(x), y) times(x, y) -> timesIter(0, x, y, 0) timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z) ifTimes(true, i, x, y, z) -> z ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y)) isZero(0) -> true isZero(s(0)) -> false isZero(s(s(x))) -> isZero(s(x)) inc(0) -> s(0) inc(s(x)) -> s(inc(x)) inc(x) -> s(x) p(0) -> 0 p(s(x)) -> x p(s(s(x))) -> s(p(s(x))) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) f0(0, y, x) -> f1(x, y, x) f1(x, y, z) -> f2(x, y, z) f2(x, 1, z) -> f0(x, z, z) f0(x, y, z) -> d f1(x, y, z) -> c 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(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(1) -> 1 encArg(d) -> d encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ifPlus(x_1, x_2, x_3)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_timesIter(x_1, x_2, x_3, x_4)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ifTimes(x_1, x_2, x_3, x_4, x_5)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_f0(x_1, x_2, x_3)) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f1(x_1, x_2, x_3)) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f2(x_1, x_2, x_3)) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_ifPlus(x_1, x_2, x_3) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isZero(x_1) -> isZero(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_timesIter(x_1, x_2, x_3, x_4) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_ifTimes(x_1, x_2, x_3, x_4, x_5) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_f0(x_1, x_2, x_3) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f1(x_1, x_2, x_3) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f2(x_1, x_2, x_3) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_1 -> 1 encode_d -> d encode_c -> c ---------------------------------------- (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: plus(x, y) -> ifPlus(isZero(x), x, inc(y)) ifPlus(true, x, y) -> p(y) ifPlus(false, x, y) -> plus(p(x), y) times(x, y) -> timesIter(0, x, y, 0) timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z) ifTimes(true, i, x, y, z) -> z ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y)) isZero(0) -> true isZero(s(0)) -> false isZero(s(s(x))) -> isZero(s(x)) inc(0) -> s(0) inc(s(x)) -> s(inc(x)) inc(x) -> s(x) p(0) -> 0 p(s(x)) -> x p(s(s(x))) -> s(p(s(x))) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) f0(0, y, x) -> f1(x, y, x) f1(x, y, z) -> f2(x, y, z) f2(x, 1, z) -> f0(x, z, z) f0(x, y, z) -> d f1(x, y, z) -> c The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(1) -> 1 encArg(d) -> d encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ifPlus(x_1, x_2, x_3)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_timesIter(x_1, x_2, x_3, x_4)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ifTimes(x_1, x_2, x_3, x_4, x_5)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_f0(x_1, x_2, x_3)) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f1(x_1, x_2, x_3)) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f2(x_1, x_2, x_3)) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_ifPlus(x_1, x_2, x_3) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isZero(x_1) -> isZero(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_timesIter(x_1, x_2, x_3, x_4) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_ifTimes(x_1, x_2, x_3, x_4, x_5) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_f0(x_1, x_2, x_3) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f1(x_1, x_2, x_3) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f2(x_1, x_2, x_3) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_1 -> 1 encode_d -> d encode_c -> c 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: plus(x, y) -> ifPlus(isZero(x), x, inc(y)) ifPlus(true, x, y) -> p(y) ifPlus(false, x, y) -> plus(p(x), y) times(x, y) -> timesIter(0, x, y, 0) timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z) ifTimes(true, i, x, y, z) -> z ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y)) isZero(0) -> true isZero(s(0)) -> false isZero(s(s(x))) -> isZero(s(x)) inc(0) -> s(0) inc(s(x)) -> s(inc(x)) inc(x) -> s(x) p(0) -> 0 p(s(x)) -> x p(s(s(x))) -> s(p(s(x))) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) f0(0, y, x) -> f1(x, y, x) f1(x, y, z) -> f2(x, y, z) f2(x, 1, z) -> f0(x, z, z) f0(x, y, z) -> d f1(x, y, z) -> c The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(1) -> 1 encArg(d) -> d encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ifPlus(x_1, x_2, x_3)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_timesIter(x_1, x_2, x_3, x_4)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ifTimes(x_1, x_2, x_3, x_4, x_5)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_f0(x_1, x_2, x_3)) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f1(x_1, x_2, x_3)) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f2(x_1, x_2, x_3)) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_ifPlus(x_1, x_2, x_3) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isZero(x_1) -> isZero(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_timesIter(x_1, x_2, x_3, x_4) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_ifTimes(x_1, x_2, x_3, x_4, x_5) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_f0(x_1, x_2, x_3) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f1(x_1, x_2, x_3) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f2(x_1, x_2, x_3) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_1 -> 1 encode_d -> d encode_c -> c 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: plus(x, y) -> ifPlus(isZero(x), x, inc(y)) ifPlus(true, x, y) -> p(y) ifPlus(false, x, y) -> plus(p(x), y) times(x, y) -> timesIter(0, x, y, 0) timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z) ifTimes(true, i, x, y, z) -> z ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y)) isZero(0) -> true isZero(s(0)) -> false isZero(s(s(x))) -> isZero(s(x)) inc(0) -> s(0) inc(s(x)) -> s(inc(x)) inc(x) -> s(x) p(0) -> 0 p(s(x)) -> x p(s(s(x))) -> s(p(s(x))) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) f0(0, y, x) -> f1(x, y, x) f1(x, y, z) -> f2(x, y, z) f2(x, 1, z) -> f0(x, z, z) f0(x, y, z) -> d f1(x, y, z) -> c The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(1) -> 1 encArg(d) -> d encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ifPlus(x_1, x_2, x_3)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_timesIter(x_1, x_2, x_3, x_4)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ifTimes(x_1, x_2, x_3, x_4, x_5)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_f0(x_1, x_2, x_3)) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f1(x_1, x_2, x_3)) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f2(x_1, x_2, x_3)) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_ifPlus(x_1, x_2, x_3) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isZero(x_1) -> isZero(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_timesIter(x_1, x_2, x_3, x_4) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_ifTimes(x_1, x_2, x_3, x_4, x_5) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_f0(x_1, x_2, x_3) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f1(x_1, x_2, x_3) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f2(x_1, x_2, x_3) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_1 -> 1 encode_d -> d encode_c -> c 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 isZero(s(s(x))) ->^+ isZero(s(x)) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(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: plus(x, y) -> ifPlus(isZero(x), x, inc(y)) ifPlus(true, x, y) -> p(y) ifPlus(false, x, y) -> plus(p(x), y) times(x, y) -> timesIter(0, x, y, 0) timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z) ifTimes(true, i, x, y, z) -> z ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y)) isZero(0) -> true isZero(s(0)) -> false isZero(s(s(x))) -> isZero(s(x)) inc(0) -> s(0) inc(s(x)) -> s(inc(x)) inc(x) -> s(x) p(0) -> 0 p(s(x)) -> x p(s(s(x))) -> s(p(s(x))) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) f0(0, y, x) -> f1(x, y, x) f1(x, y, z) -> f2(x, y, z) f2(x, 1, z) -> f0(x, z, z) f0(x, y, z) -> d f1(x, y, z) -> c The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(1) -> 1 encArg(d) -> d encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ifPlus(x_1, x_2, x_3)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_timesIter(x_1, x_2, x_3, x_4)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ifTimes(x_1, x_2, x_3, x_4, x_5)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_f0(x_1, x_2, x_3)) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f1(x_1, x_2, x_3)) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f2(x_1, x_2, x_3)) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_ifPlus(x_1, x_2, x_3) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isZero(x_1) -> isZero(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_timesIter(x_1, x_2, x_3, x_4) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_ifTimes(x_1, x_2, x_3, x_4, x_5) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_f0(x_1, x_2, x_3) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f1(x_1, x_2, x_3) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f2(x_1, x_2, x_3) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_1 -> 1 encode_d -> d encode_c -> c 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: plus(x, y) -> ifPlus(isZero(x), x, inc(y)) ifPlus(true, x, y) -> p(y) ifPlus(false, x, y) -> plus(p(x), y) times(x, y) -> timesIter(0, x, y, 0) timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z) ifTimes(true, i, x, y, z) -> z ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y)) isZero(0) -> true isZero(s(0)) -> false isZero(s(s(x))) -> isZero(s(x)) inc(0) -> s(0) inc(s(x)) -> s(inc(x)) inc(x) -> s(x) p(0) -> 0 p(s(x)) -> x p(s(s(x))) -> s(p(s(x))) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) f0(0, y, x) -> f1(x, y, x) f1(x, y, z) -> f2(x, y, z) f2(x, 1, z) -> f0(x, z, z) f0(x, y, z) -> d f1(x, y, z) -> c The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(1) -> 1 encArg(d) -> d encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ifPlus(x_1, x_2, x_3)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_timesIter(x_1, x_2, x_3, x_4)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ifTimes(x_1, x_2, x_3, x_4, x_5)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_f0(x_1, x_2, x_3)) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f1(x_1, x_2, x_3)) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_f2(x_1, x_2, x_3)) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_ifPlus(x_1, x_2, x_3) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isZero(x_1) -> isZero(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_timesIter(x_1, x_2, x_3, x_4) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_ifTimes(x_1, x_2, x_3, x_4, x_5) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_f0(x_1, x_2, x_3) -> f0(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f1(x_1, x_2, x_3) -> f1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f2(x_1, x_2, x_3) -> f2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_1 -> 1 encode_d -> d encode_c -> c Rewrite Strategy: INNERMOST