/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 521 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 6 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 (13) DecreasingLoopProof [FINISHED, 0 ms] (14) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: sort(l) -> st(0, l) st(n, l) -> cond1(member(n, l), n, l) cond1(true, n, l) -> cons(n, st(s(n), l)) cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) cond2(true, n, l) -> nil cond2(false, n, l) -> st(s(n), l) member(n, nil) -> false member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) or(x, true) -> true or(x, false) -> x equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) max(nil) -> 0 max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) if(true, u, v) -> u if(false, u, v) -> 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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(nil) -> nil encArg(cons_sort(x_1)) -> sort(encArg(x_1)) encArg(cons_st(x_1, x_2)) -> st(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_member(x_1, x_2)) -> member(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1)) -> max(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_sort(x_1) -> sort(encArg(x_1)) encode_st(x_1, x_2) -> st(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_member(x_1, x_2) -> member(encArg(x_1), encArg(x_2)) encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_max(x_1) -> max(encArg(x_1)) encode_nil -> nil encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: sort(l) -> st(0, l) st(n, l) -> cond1(member(n, l), n, l) cond1(true, n, l) -> cons(n, st(s(n), l)) cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) cond2(true, n, l) -> nil cond2(false, n, l) -> st(s(n), l) member(n, nil) -> false member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) or(x, true) -> true or(x, false) -> x equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) max(nil) -> 0 max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) if(true, u, v) -> u if(false, u, v) -> v The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(nil) -> nil encArg(cons_sort(x_1)) -> sort(encArg(x_1)) encArg(cons_st(x_1, x_2)) -> st(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_member(x_1, x_2)) -> member(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1)) -> max(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_sort(x_1) -> sort(encArg(x_1)) encode_st(x_1, x_2) -> st(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_member(x_1, x_2) -> member(encArg(x_1), encArg(x_2)) encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_max(x_1) -> max(encArg(x_1)) encode_nil -> nil encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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(EXP, INF). The TRS R consists of the following rules: sort(l) -> st(0, l) st(n, l) -> cond1(member(n, l), n, l) cond1(true, n, l) -> cons(n, st(s(n), l)) cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) cond2(true, n, l) -> nil cond2(false, n, l) -> st(s(n), l) member(n, nil) -> false member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) or(x, true) -> true or(x, false) -> x equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) max(nil) -> 0 max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) if(true, u, v) -> u if(false, u, v) -> v The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(nil) -> nil encArg(cons_sort(x_1)) -> sort(encArg(x_1)) encArg(cons_st(x_1, x_2)) -> st(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_member(x_1, x_2)) -> member(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1)) -> max(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_sort(x_1) -> sort(encArg(x_1)) encode_st(x_1, x_2) -> st(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_member(x_1, x_2) -> member(encArg(x_1), encArg(x_2)) encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_max(x_1) -> max(encArg(x_1)) encode_nil -> nil encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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(EXP, INF). The TRS R consists of the following rules: sort(l) -> st(0, l) st(n, l) -> cond1(member(n, l), n, l) cond1(true, n, l) -> cons(n, st(s(n), l)) cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) cond2(true, n, l) -> nil cond2(false, n, l) -> st(s(n), l) member(n, nil) -> false member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) or(x, true) -> true or(x, false) -> x equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) max(nil) -> 0 max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) if(true, u, v) -> u if(false, u, v) -> v The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(nil) -> nil encArg(cons_sort(x_1)) -> sort(encArg(x_1)) encArg(cons_st(x_1, x_2)) -> st(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_member(x_1, x_2)) -> member(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1)) -> max(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_sort(x_1) -> sort(encArg(x_1)) encode_st(x_1, x_2) -> st(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_member(x_1, x_2) -> member(encArg(x_1), encArg(x_2)) encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_max(x_1) -> max(encArg(x_1)) encode_nil -> nil encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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 gt(s(u), s(v)) ->^+ gt(u, v) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [u / s(u), v / s(v)]. 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(EXP, INF). The TRS R consists of the following rules: sort(l) -> st(0, l) st(n, l) -> cond1(member(n, l), n, l) cond1(true, n, l) -> cons(n, st(s(n), l)) cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) cond2(true, n, l) -> nil cond2(false, n, l) -> st(s(n), l) member(n, nil) -> false member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) or(x, true) -> true or(x, false) -> x equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) max(nil) -> 0 max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) if(true, u, v) -> u if(false, u, v) -> v The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(nil) -> nil encArg(cons_sort(x_1)) -> sort(encArg(x_1)) encArg(cons_st(x_1, x_2)) -> st(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_member(x_1, x_2)) -> member(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1)) -> max(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_sort(x_1) -> sort(encArg(x_1)) encode_st(x_1, x_2) -> st(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_member(x_1, x_2) -> member(encArg(x_1), encArg(x_2)) encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_max(x_1) -> max(encArg(x_1)) encode_nil -> nil encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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(EXP, INF). The TRS R consists of the following rules: sort(l) -> st(0, l) st(n, l) -> cond1(member(n, l), n, l) cond1(true, n, l) -> cons(n, st(s(n), l)) cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) cond2(true, n, l) -> nil cond2(false, n, l) -> st(s(n), l) member(n, nil) -> false member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) or(x, true) -> true or(x, false) -> x equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) max(nil) -> 0 max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) if(true, u, v) -> u if(false, u, v) -> v The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(nil) -> nil encArg(cons_sort(x_1)) -> sort(encArg(x_1)) encArg(cons_st(x_1, x_2)) -> st(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_member(x_1, x_2)) -> member(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1)) -> max(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_sort(x_1) -> sort(encArg(x_1)) encode_st(x_1, x_2) -> st(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_member(x_1, x_2) -> member(encArg(x_1), encArg(x_2)) encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_max(x_1) -> max(encArg(x_1)) encode_nil -> nil encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence max(cons(u, l)) ->^+ if(gt(u, max(l)), u, max(l)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1]. The pumping substitution is [l / cons(u, l)]. The result substitution is [ ]. The rewrite sequence max(cons(u, l)) ->^+ if(gt(u, max(l)), u, max(l)) gives rise to a decreasing loop by considering the right hand sides subterm at position [2]. The pumping substitution is [l / cons(u, l)]. The result substitution is [ ]. ---------------------------------------- (14) BOUNDS(EXP, INF)