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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 337 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) quot(x, 0) -> quotZeroErro quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) if(true, x, y, z, u, v) -> v if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) 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(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(quotZeroErro) -> quotZeroErro encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_quotIter(x_1, x_2, x_3, x_4, x_5)) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_quotZeroErro -> quotZeroErro encode_quotIter(x_1, x_2, x_3, x_4, x_5) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) ---------------------------------------- (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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) quot(x, 0) -> quotZeroErro quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) if(true, x, y, z, u, v) -> v if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(quotZeroErro) -> quotZeroErro encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_quotIter(x_1, x_2, x_3, x_4, x_5)) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_quotZeroErro -> quotZeroErro encode_quotIter(x_1, x_2, x_3, x_4, x_5) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) quot(x, 0) -> quotZeroErro quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) if(true, x, y, z, u, v) -> v if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(quotZeroErro) -> quotZeroErro encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_quotIter(x_1, x_2, x_3, x_4, x_5)) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_quotZeroErro -> quotZeroErro encode_quotIter(x_1, x_2, x_3, x_4, x_5) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) quot(x, 0) -> quotZeroErro quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) if(true, x, y, z, u, v) -> v if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(quotZeroErro) -> quotZeroErro encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_quotIter(x_1, x_2, x_3, x_4, x_5)) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_quotZeroErro -> quotZeroErro encode_quotIter(x_1, x_2, x_3, x_4, x_5) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) 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 le(s(x), s(y)) ->^+ le(x, y) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x), y / s(y)]. 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) quot(x, 0) -> quotZeroErro quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) if(true, x, y, z, u, v) -> v if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(quotZeroErro) -> quotZeroErro encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_quotIter(x_1, x_2, x_3, x_4, x_5)) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_quotZeroErro -> quotZeroErro encode_quotIter(x_1, x_2, x_3, x_4, x_5) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) quot(x, 0) -> quotZeroErro quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) if(true, x, y, z, u, v) -> v if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(quotZeroErro) -> quotZeroErro encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_quotIter(x_1, x_2, x_3, x_4, x_5)) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_quotZeroErro -> quotZeroErro encode_quotIter(x_1, x_2, x_3, x_4, x_5) -> quotIter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) Rewrite Strategy: INNERMOST