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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 520 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: lessElements(l, t) -> lessE(l, t, 0) lessE(l, t, n) -> if(le(length(l), n), le(length(toList(t)), n), l, t, n) if(true, b, l, t, n) -> l if(false, true, l, t, n) -> t if(false, false, l, t, n) -> lessE(l, t, s(n)) length(nil) -> 0 length(cons(n, l)) -> s(length(l)) toList(leaf) -> nil toList(node(t1, n, t2)) -> append(toList(t1), cons(n, toList(t2))) append(nil, l2) -> l2 append(cons(n, l1), l2) -> cons(n, append(l1, l2)) le(s(n), 0) -> false le(0, m) -> true le(s(n), s(m)) -> le(n, m) a -> c a -> d 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(leaf) -> leaf encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(c) -> c encArg(d) -> d encArg(cons_lessElements(x_1, x_2)) -> lessElements(encArg(x_1), encArg(x_2)) encArg(cons_lessE(x_1, x_2, x_3)) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) 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_length(x_1)) -> length(encArg(x_1)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_lessElements(x_1, x_2) -> lessElements(encArg(x_1), encArg(x_2)) encode_lessE(x_1, x_2, x_3) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 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_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_toList(x_1) -> toList(encArg(x_1)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_leaf -> leaf encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_a -> a encode_c -> c encode_d -> d ---------------------------------------- (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: lessElements(l, t) -> lessE(l, t, 0) lessE(l, t, n) -> if(le(length(l), n), le(length(toList(t)), n), l, t, n) if(true, b, l, t, n) -> l if(false, true, l, t, n) -> t if(false, false, l, t, n) -> lessE(l, t, s(n)) length(nil) -> 0 length(cons(n, l)) -> s(length(l)) toList(leaf) -> nil toList(node(t1, n, t2)) -> append(toList(t1), cons(n, toList(t2))) append(nil, l2) -> l2 append(cons(n, l1), l2) -> cons(n, append(l1, l2)) le(s(n), 0) -> false le(0, m) -> true le(s(n), s(m)) -> le(n, m) a -> c a -> d 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(leaf) -> leaf encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(c) -> c encArg(d) -> d encArg(cons_lessElements(x_1, x_2)) -> lessElements(encArg(x_1), encArg(x_2)) encArg(cons_lessE(x_1, x_2, x_3)) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) 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_length(x_1)) -> length(encArg(x_1)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_lessElements(x_1, x_2) -> lessElements(encArg(x_1), encArg(x_2)) encode_lessE(x_1, x_2, x_3) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 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_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_toList(x_1) -> toList(encArg(x_1)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_leaf -> leaf encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_a -> a encode_c -> c encode_d -> d 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: lessElements(l, t) -> lessE(l, t, 0) lessE(l, t, n) -> if(le(length(l), n), le(length(toList(t)), n), l, t, n) if(true, b, l, t, n) -> l if(false, true, l, t, n) -> t if(false, false, l, t, n) -> lessE(l, t, s(n)) length(nil) -> 0 length(cons(n, l)) -> s(length(l)) toList(leaf) -> nil toList(node(t1, n, t2)) -> append(toList(t1), cons(n, toList(t2))) append(nil, l2) -> l2 append(cons(n, l1), l2) -> cons(n, append(l1, l2)) le(s(n), 0) -> false le(0, m) -> true le(s(n), s(m)) -> le(n, m) a -> c a -> d 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(leaf) -> leaf encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(c) -> c encArg(d) -> d encArg(cons_lessElements(x_1, x_2)) -> lessElements(encArg(x_1), encArg(x_2)) encArg(cons_lessE(x_1, x_2, x_3)) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) 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_length(x_1)) -> length(encArg(x_1)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_lessElements(x_1, x_2) -> lessElements(encArg(x_1), encArg(x_2)) encode_lessE(x_1, x_2, x_3) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 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_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_toList(x_1) -> toList(encArg(x_1)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_leaf -> leaf encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_a -> a encode_c -> c encode_d -> d 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: lessElements(l, t) -> lessE(l, t, 0) lessE(l, t, n) -> if(le(length(l), n), le(length(toList(t)), n), l, t, n) if(true, b, l, t, n) -> l if(false, true, l, t, n) -> t if(false, false, l, t, n) -> lessE(l, t, s(n)) length(nil) -> 0 length(cons(n, l)) -> s(length(l)) toList(leaf) -> nil toList(node(t1, n, t2)) -> append(toList(t1), cons(n, toList(t2))) append(nil, l2) -> l2 append(cons(n, l1), l2) -> cons(n, append(l1, l2)) le(s(n), 0) -> false le(0, m) -> true le(s(n), s(m)) -> le(n, m) a -> c a -> d 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(leaf) -> leaf encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(c) -> c encArg(d) -> d encArg(cons_lessElements(x_1, x_2)) -> lessElements(encArg(x_1), encArg(x_2)) encArg(cons_lessE(x_1, x_2, x_3)) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) 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_length(x_1)) -> length(encArg(x_1)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_lessElements(x_1, x_2) -> lessElements(encArg(x_1), encArg(x_2)) encode_lessE(x_1, x_2, x_3) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 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_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_toList(x_1) -> toList(encArg(x_1)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_leaf -> leaf encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_a -> a encode_c -> c encode_d -> d 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 toList(node(t1, n, t2)) ->^+ append(toList(t1), cons(n, toList(t2))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [t1 / node(t1, n, t2)]. 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: lessElements(l, t) -> lessE(l, t, 0) lessE(l, t, n) -> if(le(length(l), n), le(length(toList(t)), n), l, t, n) if(true, b, l, t, n) -> l if(false, true, l, t, n) -> t if(false, false, l, t, n) -> lessE(l, t, s(n)) length(nil) -> 0 length(cons(n, l)) -> s(length(l)) toList(leaf) -> nil toList(node(t1, n, t2)) -> append(toList(t1), cons(n, toList(t2))) append(nil, l2) -> l2 append(cons(n, l1), l2) -> cons(n, append(l1, l2)) le(s(n), 0) -> false le(0, m) -> true le(s(n), s(m)) -> le(n, m) a -> c a -> d 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(leaf) -> leaf encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(c) -> c encArg(d) -> d encArg(cons_lessElements(x_1, x_2)) -> lessElements(encArg(x_1), encArg(x_2)) encArg(cons_lessE(x_1, x_2, x_3)) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) 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_length(x_1)) -> length(encArg(x_1)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_lessElements(x_1, x_2) -> lessElements(encArg(x_1), encArg(x_2)) encode_lessE(x_1, x_2, x_3) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 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_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_toList(x_1) -> toList(encArg(x_1)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_leaf -> leaf encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_a -> a encode_c -> c encode_d -> d 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: lessElements(l, t) -> lessE(l, t, 0) lessE(l, t, n) -> if(le(length(l), n), le(length(toList(t)), n), l, t, n) if(true, b, l, t, n) -> l if(false, true, l, t, n) -> t if(false, false, l, t, n) -> lessE(l, t, s(n)) length(nil) -> 0 length(cons(n, l)) -> s(length(l)) toList(leaf) -> nil toList(node(t1, n, t2)) -> append(toList(t1), cons(n, toList(t2))) append(nil, l2) -> l2 append(cons(n, l1), l2) -> cons(n, append(l1, l2)) le(s(n), 0) -> false le(0, m) -> true le(s(n), s(m)) -> le(n, m) a -> c a -> d 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(leaf) -> leaf encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(c) -> c encArg(d) -> d encArg(cons_lessElements(x_1, x_2)) -> lessElements(encArg(x_1), encArg(x_2)) encArg(cons_lessE(x_1, x_2, x_3)) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) 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_length(x_1)) -> length(encArg(x_1)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_lessElements(x_1, x_2) -> lessElements(encArg(x_1), encArg(x_2)) encode_lessE(x_1, x_2, x_3) -> lessE(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 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_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_toList(x_1) -> toList(encArg(x_1)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_leaf -> leaf encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_a -> a encode_c -> c encode_d -> d Rewrite Strategy: INNERMOST