/export/starexec/sandbox2/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 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:

   active(incr(nil)) -> mark(nil)
   active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
   active(adx(nil)) -> mark(nil)
   active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
   active(nats) -> mark(adx(zeros))
   active(zeros) -> mark(cons(0, zeros))
   active(head(cons(X, L))) -> mark(X)
   active(tail(cons(X, L))) -> mark(L)
   active(incr(X)) -> incr(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(adx(X)) -> adx(active(X))
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   incr(mark(X)) -> mark(incr(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   adx(mark(X)) -> mark(adx(X))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   proper(incr(X)) -> incr(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(adx(X)) -> adx(proper(X))
   proper(nats) -> ok(nats)
   proper(zeros) -> ok(zeros)
   proper(0) -> ok(0)
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   incr(ok(X)) -> ok(incr(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   adx(ok(X)) -> ok(adx(X))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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:

   active(incr(nil)) -> mark(nil)
   active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
   active(adx(nil)) -> mark(nil)
   active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
   active(nats) -> mark(adx(zeros))
   active(zeros) -> mark(cons(0, zeros))
   active(head(cons(X, L))) -> mark(X)
   active(tail(cons(X, L))) -> mark(L)
   active(incr(X)) -> incr(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(adx(X)) -> adx(active(X))
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   incr(mark(X)) -> mark(incr(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   adx(mark(X)) -> mark(adx(X))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   proper(incr(X)) -> incr(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(adx(X)) -> adx(proper(X))
   proper(nats) -> ok(nats)
   proper(zeros) -> ok(zeros)
   proper(0) -> ok(0)
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   incr(ok(X)) -> ok(incr(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   adx(ok(X)) -> ok(adx(X))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

head(ok(X)) ->^+ ok(head(X))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X / ok(X)].

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:

   active(incr(nil)) -> mark(nil)
   active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
   active(adx(nil)) -> mark(nil)
   active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
   active(nats) -> mark(adx(zeros))
   active(zeros) -> mark(cons(0, zeros))
   active(head(cons(X, L))) -> mark(X)
   active(tail(cons(X, L))) -> mark(L)
   active(incr(X)) -> incr(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(adx(X)) -> adx(active(X))
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   incr(mark(X)) -> mark(incr(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   adx(mark(X)) -> mark(adx(X))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   proper(incr(X)) -> incr(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(adx(X)) -> adx(proper(X))
   proper(nats) -> ok(nats)
   proper(zeros) -> ok(zeros)
   proper(0) -> ok(0)
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   incr(ok(X)) -> ok(incr(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   adx(ok(X)) -> ok(adx(X))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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:

   active(incr(nil)) -> mark(nil)
   active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
   active(adx(nil)) -> mark(nil)
   active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
   active(nats) -> mark(adx(zeros))
   active(zeros) -> mark(cons(0, zeros))
   active(head(cons(X, L))) -> mark(X)
   active(tail(cons(X, L))) -> mark(L)
   active(incr(X)) -> incr(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(adx(X)) -> adx(active(X))
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   incr(mark(X)) -> mark(incr(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   adx(mark(X)) -> mark(adx(X))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   proper(incr(X)) -> incr(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(adx(X)) -> adx(proper(X))
   proper(nats) -> ok(nats)
   proper(zeros) -> ok(zeros)
   proper(0) -> ok(0)
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   incr(ok(X)) -> ok(incr(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   adx(ok(X)) -> ok(adx(X))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL