/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 845 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 4 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: lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) if(true, x, y, z, u) -> z if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) if2(true, x, y, z, u) -> z if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(x, y) -> ifTimes(ge(0, x), x, y) ifTimes(true, x, y) -> 0 ifTimes(false, x, y) -> plus(y, times(y, p(x))) p(s(x)) -> x p(0) -> s(s(0)) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) or(true, y) -> true or(false, y) -> y divisible(0, s(y)) -> true divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) div(x, y, 0) -> divisible(x, y) div(0, y, s(z)) -> false div(s(x), y, s(z)) -> div(x, y, z) a -> b a -> 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(0) -> 0 encArg(true) -> true encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_lcm(x_1, x_2)) -> lcm(encArg(x_1), encArg(x_2)) encArg(cons_lcmIter(x_1, x_2, x_3, x_4)) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_ifTimes(x_1, x_2, x_3)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) 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_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_divisible(x_1, x_2)) -> divisible(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a) -> a encode_lcm(x_1, x_2) -> lcm(encArg(x_1), encArg(x_2)) encode_lcmIter(x_1, x_2, x_3, x_4) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_divisible(x_1, x_2) -> divisible(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_ifTimes(x_1, x_2, x_3) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_b -> b 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: lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) if(true, x, y, z, u) -> z if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) if2(true, x, y, z, u) -> z if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(x, y) -> ifTimes(ge(0, x), x, y) ifTimes(true, x, y) -> 0 ifTimes(false, x, y) -> plus(y, times(y, p(x))) p(s(x)) -> x p(0) -> s(s(0)) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) or(true, y) -> true or(false, y) -> y divisible(0, s(y)) -> true divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) div(x, y, 0) -> divisible(x, y) div(0, y, s(z)) -> false div(s(x), y, s(z)) -> div(x, y, z) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_lcm(x_1, x_2)) -> lcm(encArg(x_1), encArg(x_2)) encArg(cons_lcmIter(x_1, x_2, x_3, x_4)) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_ifTimes(x_1, x_2, x_3)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) 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_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_divisible(x_1, x_2)) -> divisible(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a) -> a encode_lcm(x_1, x_2) -> lcm(encArg(x_1), encArg(x_2)) encode_lcmIter(x_1, x_2, x_3, x_4) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_divisible(x_1, x_2) -> divisible(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_ifTimes(x_1, x_2, x_3) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_b -> b 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: lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) if(true, x, y, z, u) -> z if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) if2(true, x, y, z, u) -> z if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(x, y) -> ifTimes(ge(0, x), x, y) ifTimes(true, x, y) -> 0 ifTimes(false, x, y) -> plus(y, times(y, p(x))) p(s(x)) -> x p(0) -> s(s(0)) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) or(true, y) -> true or(false, y) -> y divisible(0, s(y)) -> true divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) div(x, y, 0) -> divisible(x, y) div(0, y, s(z)) -> false div(s(x), y, s(z)) -> div(x, y, z) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_lcm(x_1, x_2)) -> lcm(encArg(x_1), encArg(x_2)) encArg(cons_lcmIter(x_1, x_2, x_3, x_4)) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_ifTimes(x_1, x_2, x_3)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) 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_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_divisible(x_1, x_2)) -> divisible(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a) -> a encode_lcm(x_1, x_2) -> lcm(encArg(x_1), encArg(x_2)) encode_lcmIter(x_1, x_2, x_3, x_4) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_divisible(x_1, x_2) -> divisible(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_ifTimes(x_1, x_2, x_3) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_b -> b 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: lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) if(true, x, y, z, u) -> z if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) if2(true, x, y, z, u) -> z if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(x, y) -> ifTimes(ge(0, x), x, y) ifTimes(true, x, y) -> 0 ifTimes(false, x, y) -> plus(y, times(y, p(x))) p(s(x)) -> x p(0) -> s(s(0)) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) or(true, y) -> true or(false, y) -> y divisible(0, s(y)) -> true divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) div(x, y, 0) -> divisible(x, y) div(0, y, s(z)) -> false div(s(x), y, s(z)) -> div(x, y, z) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_lcm(x_1, x_2)) -> lcm(encArg(x_1), encArg(x_2)) encArg(cons_lcmIter(x_1, x_2, x_3, x_4)) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_ifTimes(x_1, x_2, x_3)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) 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_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_divisible(x_1, x_2)) -> divisible(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a) -> a encode_lcm(x_1, x_2) -> lcm(encArg(x_1), encArg(x_2)) encode_lcmIter(x_1, x_2, x_3, x_4) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_divisible(x_1, x_2) -> divisible(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_ifTimes(x_1, x_2, x_3) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_b -> b 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 div(s(x), y, s(z)) ->^+ div(x, y, z) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x), z / s(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(n^1, INF). The TRS R consists of the following rules: lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) if(true, x, y, z, u) -> z if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) if2(true, x, y, z, u) -> z if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(x, y) -> ifTimes(ge(0, x), x, y) ifTimes(true, x, y) -> 0 ifTimes(false, x, y) -> plus(y, times(y, p(x))) p(s(x)) -> x p(0) -> s(s(0)) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) or(true, y) -> true or(false, y) -> y divisible(0, s(y)) -> true divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) div(x, y, 0) -> divisible(x, y) div(0, y, s(z)) -> false div(s(x), y, s(z)) -> div(x, y, z) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_lcm(x_1, x_2)) -> lcm(encArg(x_1), encArg(x_2)) encArg(cons_lcmIter(x_1, x_2, x_3, x_4)) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_ifTimes(x_1, x_2, x_3)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) 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_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_divisible(x_1, x_2)) -> divisible(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a) -> a encode_lcm(x_1, x_2) -> lcm(encArg(x_1), encArg(x_2)) encode_lcmIter(x_1, x_2, x_3, x_4) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_divisible(x_1, x_2) -> divisible(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_ifTimes(x_1, x_2, x_3) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_b -> b 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: lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) if(true, x, y, z, u) -> z if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) if2(true, x, y, z, u) -> z if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(x, y) -> ifTimes(ge(0, x), x, y) ifTimes(true, x, y) -> 0 ifTimes(false, x, y) -> plus(y, times(y, p(x))) p(s(x)) -> x p(0) -> s(s(0)) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) or(true, y) -> true or(false, y) -> y divisible(0, s(y)) -> true divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) div(x, y, 0) -> divisible(x, y) div(0, y, s(z)) -> false div(s(x), y, s(z)) -> div(x, y, z) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_lcm(x_1, x_2)) -> lcm(encArg(x_1), encArg(x_2)) encArg(cons_lcmIter(x_1, x_2, x_3, x_4)) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_ifTimes(x_1, x_2, x_3)) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) 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_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_divisible(x_1, x_2)) -> divisible(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a) -> a encode_lcm(x_1, x_2) -> lcm(encArg(x_1), encArg(x_2)) encode_lcmIter(x_1, x_2, x_3, x_4) -> lcmIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_divisible(x_1, x_2) -> divisible(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_ifTimes(x_1, x_2, x_3) -> ifTimes(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST