/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) head(cons(h, t)) -> h tail(cons(h, t)) -> t ring(st_1, in_2, st_2, in_3, st_3, m) -> if_1(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_1))) if_1(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_2(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_2))) if_2(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_2))) if_3(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m) if_2(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(two, head(in_2)), st_2)))) if_4(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_5(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(two, head(in_2)))) if_5(st_1, in_2, st_2, in_3, st_3, m, true) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_6(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_3))) if_6(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_3))) if_7(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) if_6(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(three, head(in_3)), st_3)))) if_8(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_9(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(three, head(in_3)))) if_9(st_1, in_2, st_2, in_3, st_3, m, true) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) head(cons(h, t)) -> h tail(cons(h, t)) -> t ring(st_1, in_2, st_2, in_3, st_3, m) -> if_1(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_1))) if_1(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_2(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_2))) if_2(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_2))) if_3(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m) if_2(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(two, head(in_2)), st_2)))) if_4(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_5(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(two, head(in_2)))) if_5(st_1, in_2, st_2, in_3, st_3, m, true) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_6(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_3))) if_6(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_3))) if_7(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) if_6(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(three, head(in_3)), st_3)))) if_8(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_9(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(three, head(in_3)))) if_9(st_1, in_2, st_2, in_3, st_3, m, true) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence map_f(pid, cons(h, t)) ->^+ app(f(pid, h), map_f(pid, t)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [t / cons(h, t)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) head(cons(h, t)) -> h tail(cons(h, t)) -> t ring(st_1, in_2, st_2, in_3, st_3, m) -> if_1(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_1))) if_1(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_2(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_2))) if_2(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_2))) if_3(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m) if_2(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(two, head(in_2)), st_2)))) if_4(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_5(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(two, head(in_2)))) if_5(st_1, in_2, st_2, in_3, st_3, m, true) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_6(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_3))) if_6(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_3))) if_7(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) if_6(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(three, head(in_3)), st_3)))) if_8(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_9(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(three, head(in_3)))) if_9(st_1, in_2, st_2, in_3, st_3, m, true) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m) S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) head(cons(h, t)) -> h tail(cons(h, t)) -> t ring(st_1, in_2, st_2, in_3, st_3, m) -> if_1(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_1))) if_1(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_2(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_2))) if_2(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_2))) if_3(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m) if_2(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(two, head(in_2)), st_2)))) if_4(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_5(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(two, head(in_2)))) if_5(st_1, in_2, st_2, in_3, st_3, m, true) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_6(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_3))) if_6(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_3))) if_7(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) if_6(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(three, head(in_3)), st_3)))) if_8(st_1, in_2, st_2, in_3, st_3, m, false) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) ring(st_1, in_2, st_2, in_3, st_3, m) -> if_9(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(three, head(in_3)))) if_9(st_1, in_2, st_2, in_3, st_3, m, true) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m) S is empty. Rewrite Strategy: FULL