/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 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), 313 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 4 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:

   a__incr(nil) -> nil
   a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L))
   a__adx(nil) -> nil
   a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L)))
   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__head(cons(X, L)) -> mark(X)
   a__tail(cons(X, L)) -> mark(L)
   mark(incr(X)) -> a__incr(mark(X))
   mark(adx(X)) -> a__adx(mark(X))
   mark(nats) -> a__nats
   mark(zeros) -> a__zeros
   mark(head(X)) -> a__head(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__incr(X) -> incr(X)
   a__adx(X) -> adx(X)
   a__nats -> nats
   a__zeros -> zeros
   a__head(X) -> head(X)
   a__tail(X) -> tail(X)

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(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(incr(x_1)) -> incr(encArg(x_1))
   encArg(adx(x_1)) -> adx(encArg(x_1))
   encArg(0) -> 0
   encArg(zeros) -> zeros
   encArg(nats) -> nats
   encArg(head(x_1)) -> head(encArg(x_1))
   encArg(tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1))
   encArg(cons_a__adx(x_1)) -> a__adx(encArg(x_1))
   encArg(cons_a__nats) -> a__nats
   encArg(cons_a__zeros) -> a__zeros
   encArg(cons_a__head(x_1)) -> a__head(encArg(x_1))
   encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__incr(x_1) -> a__incr(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_a__adx(x_1) -> a__adx(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_a__nats -> a__nats
   encode_a__zeros -> a__zeros
   encode_0 -> 0
   encode_zeros -> zeros
   encode_a__head(x_1) -> a__head(encArg(x_1))
   encode_a__tail(x_1) -> a__tail(encArg(x_1))
   encode_nats -> nats
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))


----------------------------------------

(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:

   a__incr(nil) -> nil
   a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L))
   a__adx(nil) -> nil
   a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L)))
   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__head(cons(X, L)) -> mark(X)
   a__tail(cons(X, L)) -> mark(L)
   mark(incr(X)) -> a__incr(mark(X))
   mark(adx(X)) -> a__adx(mark(X))
   mark(nats) -> a__nats
   mark(zeros) -> a__zeros
   mark(head(X)) -> a__head(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__incr(X) -> incr(X)
   a__adx(X) -> adx(X)
   a__nats -> nats
   a__zeros -> zeros
   a__head(X) -> head(X)
   a__tail(X) -> tail(X)

The (relative) TRS S consists of the following rules:

   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(incr(x_1)) -> incr(encArg(x_1))
   encArg(adx(x_1)) -> adx(encArg(x_1))
   encArg(0) -> 0
   encArg(zeros) -> zeros
   encArg(nats) -> nats
   encArg(head(x_1)) -> head(encArg(x_1))
   encArg(tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1))
   encArg(cons_a__adx(x_1)) -> a__adx(encArg(x_1))
   encArg(cons_a__nats) -> a__nats
   encArg(cons_a__zeros) -> a__zeros
   encArg(cons_a__head(x_1)) -> a__head(encArg(x_1))
   encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__incr(x_1) -> a__incr(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_a__adx(x_1) -> a__adx(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_a__nats -> a__nats
   encode_a__zeros -> a__zeros
   encode_0 -> 0
   encode_zeros -> zeros
   encode_a__head(x_1) -> a__head(encArg(x_1))
   encode_a__tail(x_1) -> a__tail(encArg(x_1))
   encode_nats -> nats
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))

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:

   a__incr(nil) -> nil
   a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L))
   a__adx(nil) -> nil
   a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L)))
   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__head(cons(X, L)) -> mark(X)
   a__tail(cons(X, L)) -> mark(L)
   mark(incr(X)) -> a__incr(mark(X))
   mark(adx(X)) -> a__adx(mark(X))
   mark(nats) -> a__nats
   mark(zeros) -> a__zeros
   mark(head(X)) -> a__head(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__incr(X) -> incr(X)
   a__adx(X) -> adx(X)
   a__nats -> nats
   a__zeros -> zeros
   a__head(X) -> head(X)
   a__tail(X) -> tail(X)

The (relative) TRS S consists of the following rules:

   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(incr(x_1)) -> incr(encArg(x_1))
   encArg(adx(x_1)) -> adx(encArg(x_1))
   encArg(0) -> 0
   encArg(zeros) -> zeros
   encArg(nats) -> nats
   encArg(head(x_1)) -> head(encArg(x_1))
   encArg(tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1))
   encArg(cons_a__adx(x_1)) -> a__adx(encArg(x_1))
   encArg(cons_a__nats) -> a__nats
   encArg(cons_a__zeros) -> a__zeros
   encArg(cons_a__head(x_1)) -> a__head(encArg(x_1))
   encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__incr(x_1) -> a__incr(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_a__adx(x_1) -> a__adx(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_a__nats -> a__nats
   encode_a__zeros -> a__zeros
   encode_0 -> 0
   encode_zeros -> zeros
   encode_a__head(x_1) -> a__head(encArg(x_1))
   encode_a__tail(x_1) -> a__tail(encArg(x_1))
   encode_nats -> nats
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))

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:

   a__incr(nil) -> nil
   a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L))
   a__adx(nil) -> nil
   a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L)))
   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__head(cons(X, L)) -> mark(X)
   a__tail(cons(X, L)) -> mark(L)
   mark(incr(X)) -> a__incr(mark(X))
   mark(adx(X)) -> a__adx(mark(X))
   mark(nats) -> a__nats
   mark(zeros) -> a__zeros
   mark(head(X)) -> a__head(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__incr(X) -> incr(X)
   a__adx(X) -> adx(X)
   a__nats -> nats
   a__zeros -> zeros
   a__head(X) -> head(X)
   a__tail(X) -> tail(X)

The (relative) TRS S consists of the following rules:

   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(incr(x_1)) -> incr(encArg(x_1))
   encArg(adx(x_1)) -> adx(encArg(x_1))
   encArg(0) -> 0
   encArg(zeros) -> zeros
   encArg(nats) -> nats
   encArg(head(x_1)) -> head(encArg(x_1))
   encArg(tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1))
   encArg(cons_a__adx(x_1)) -> a__adx(encArg(x_1))
   encArg(cons_a__nats) -> a__nats
   encArg(cons_a__zeros) -> a__zeros
   encArg(cons_a__head(x_1)) -> a__head(encArg(x_1))
   encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__incr(x_1) -> a__incr(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_a__adx(x_1) -> a__adx(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_a__nats -> a__nats
   encode_a__zeros -> a__zeros
   encode_0 -> 0
   encode_zeros -> zeros
   encode_a__head(x_1) -> a__head(encArg(x_1))
   encode_a__tail(x_1) -> a__tail(encArg(x_1))
   encode_nats -> nats
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))

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

mark(incr(X)) ->^+ a__incr(mark(X))

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

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

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:

   a__incr(nil) -> nil
   a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L))
   a__adx(nil) -> nil
   a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L)))
   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__head(cons(X, L)) -> mark(X)
   a__tail(cons(X, L)) -> mark(L)
   mark(incr(X)) -> a__incr(mark(X))
   mark(adx(X)) -> a__adx(mark(X))
   mark(nats) -> a__nats
   mark(zeros) -> a__zeros
   mark(head(X)) -> a__head(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__incr(X) -> incr(X)
   a__adx(X) -> adx(X)
   a__nats -> nats
   a__zeros -> zeros
   a__head(X) -> head(X)
   a__tail(X) -> tail(X)

The (relative) TRS S consists of the following rules:

   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(incr(x_1)) -> incr(encArg(x_1))
   encArg(adx(x_1)) -> adx(encArg(x_1))
   encArg(0) -> 0
   encArg(zeros) -> zeros
   encArg(nats) -> nats
   encArg(head(x_1)) -> head(encArg(x_1))
   encArg(tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1))
   encArg(cons_a__adx(x_1)) -> a__adx(encArg(x_1))
   encArg(cons_a__nats) -> a__nats
   encArg(cons_a__zeros) -> a__zeros
   encArg(cons_a__head(x_1)) -> a__head(encArg(x_1))
   encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__incr(x_1) -> a__incr(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_a__adx(x_1) -> a__adx(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_a__nats -> a__nats
   encode_a__zeros -> a__zeros
   encode_0 -> 0
   encode_zeros -> zeros
   encode_a__head(x_1) -> a__head(encArg(x_1))
   encode_a__tail(x_1) -> a__tail(encArg(x_1))
   encode_nats -> nats
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))

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:

   a__incr(nil) -> nil
   a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L))
   a__adx(nil) -> nil
   a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L)))
   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__head(cons(X, L)) -> mark(X)
   a__tail(cons(X, L)) -> mark(L)
   mark(incr(X)) -> a__incr(mark(X))
   mark(adx(X)) -> a__adx(mark(X))
   mark(nats) -> a__nats
   mark(zeros) -> a__zeros
   mark(head(X)) -> a__head(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__incr(X) -> incr(X)
   a__adx(X) -> adx(X)
   a__nats -> nats
   a__zeros -> zeros
   a__head(X) -> head(X)
   a__tail(X) -> tail(X)

The (relative) TRS S consists of the following rules:

   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(incr(x_1)) -> incr(encArg(x_1))
   encArg(adx(x_1)) -> adx(encArg(x_1))
   encArg(0) -> 0
   encArg(zeros) -> zeros
   encArg(nats) -> nats
   encArg(head(x_1)) -> head(encArg(x_1))
   encArg(tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_a__incr(x_1)) -> a__incr(encArg(x_1))
   encArg(cons_a__adx(x_1)) -> a__adx(encArg(x_1))
   encArg(cons_a__nats) -> a__nats
   encArg(cons_a__zeros) -> a__zeros
   encArg(cons_a__head(x_1)) -> a__head(encArg(x_1))
   encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__incr(x_1) -> a__incr(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_a__adx(x_1) -> a__adx(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_a__nats -> a__nats
   encode_a__zeros -> a__zeros
   encode_0 -> 0
   encode_zeros -> zeros
   encode_a__head(x_1) -> a__head(encArg(x_1))
   encode_a__tail(x_1) -> a__tail(encArg(x_1))
   encode_nats -> nats
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))

Rewrite Strategy: INNERMOST