/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(INF, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 343 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
        (13) InfiniteLowerBoundProof [FINISHED, 703 ms]
        (14) BOUNDS(INF, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
   nats -> adx(zeros)
   zeros -> cons(0, n__zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> activate(L)
   incr(X) -> n__incr(X)
   adx(X) -> n__adx(X)
   zeros -> n__zeros
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__adx(X)) -> adx(activate(X))
   activate(n__zeros) -> zeros
   activate(X) -> 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(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(n__adx(x_1)) -> n__adx(encArg(x_1))
   encArg(0) -> 0
   encArg(n__zeros) -> n__zeros
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_adx(x_1)) -> adx(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_zeros) -> zeros
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_incr(x_1) -> 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_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_n__adx(x_1) -> n__adx(encArg(x_1))
   encode_nats -> nats
   encode_zeros -> zeros
   encode_0 -> 0
   encode_n__zeros -> n__zeros
   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(INF, INF).


The TRS R consists of the following rules:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
   nats -> adx(zeros)
   zeros -> cons(0, n__zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> activate(L)
   incr(X) -> n__incr(X)
   adx(X) -> n__adx(X)
   zeros -> n__zeros
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__adx(X)) -> adx(activate(X))
   activate(n__zeros) -> zeros
   activate(X) -> 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(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(n__adx(x_1)) -> n__adx(encArg(x_1))
   encArg(0) -> 0
   encArg(n__zeros) -> n__zeros
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_adx(x_1)) -> adx(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_zeros) -> zeros
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_incr(x_1) -> 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_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_n__adx(x_1) -> n__adx(encArg(x_1))
   encode_nats -> nats
   encode_zeros -> zeros
   encode_0 -> 0
   encode_n__zeros -> n__zeros
   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(INF, INF).


The TRS R consists of the following rules:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
   nats -> adx(zeros)
   zeros -> cons(0, n__zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> activate(L)
   incr(X) -> n__incr(X)
   adx(X) -> n__adx(X)
   zeros -> n__zeros
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__adx(X)) -> adx(activate(X))
   activate(n__zeros) -> zeros
   activate(X) -> 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(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(n__adx(x_1)) -> n__adx(encArg(x_1))
   encArg(0) -> 0
   encArg(n__zeros) -> n__zeros
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_adx(x_1)) -> adx(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_zeros) -> zeros
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_incr(x_1) -> 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_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_n__adx(x_1) -> n__adx(encArg(x_1))
   encode_nats -> nats
   encode_zeros -> zeros
   encode_0 -> 0
   encode_n__zeros -> n__zeros
   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(INF, INF).


The TRS R consists of the following rules:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
   nats -> adx(zeros)
   zeros -> cons(0, n__zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> activate(L)
   incr(X) -> n__incr(X)
   adx(X) -> n__adx(X)
   zeros -> n__zeros
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__adx(X)) -> adx(activate(X))
   activate(n__zeros) -> zeros
   activate(X) -> 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(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(n__adx(x_1)) -> n__adx(encArg(x_1))
   encArg(0) -> 0
   encArg(n__zeros) -> n__zeros
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_adx(x_1)) -> adx(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_zeros) -> zeros
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_incr(x_1) -> 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_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_n__adx(x_1) -> n__adx(encArg(x_1))
   encode_nats -> nats
   encode_zeros -> zeros
   encode_0 -> 0
   encode_n__zeros -> n__zeros
   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

activate(n__adx(X)) ->^+ adx(activate(X))

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

The pumping substitution is [X / n__adx(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(INF, INF).


The TRS R consists of the following rules:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
   nats -> adx(zeros)
   zeros -> cons(0, n__zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> activate(L)
   incr(X) -> n__incr(X)
   adx(X) -> n__adx(X)
   zeros -> n__zeros
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__adx(X)) -> adx(activate(X))
   activate(n__zeros) -> zeros
   activate(X) -> 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(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(n__adx(x_1)) -> n__adx(encArg(x_1))
   encArg(0) -> 0
   encArg(n__zeros) -> n__zeros
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_adx(x_1)) -> adx(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_zeros) -> zeros
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_incr(x_1) -> 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_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_n__adx(x_1) -> n__adx(encArg(x_1))
   encode_nats -> nats
   encode_zeros -> zeros
   encode_0 -> 0
   encode_n__zeros -> n__zeros
   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(INF, INF).


The TRS R consists of the following rules:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
   nats -> adx(zeros)
   zeros -> cons(0, n__zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> activate(L)
   incr(X) -> n__incr(X)
   adx(X) -> n__adx(X)
   zeros -> n__zeros
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__adx(X)) -> adx(activate(X))
   activate(n__zeros) -> zeros
   activate(X) -> 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(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(n__adx(x_1)) -> n__adx(encArg(x_1))
   encArg(0) -> 0
   encArg(n__zeros) -> n__zeros
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_adx(x_1)) -> adx(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_zeros) -> zeros
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_incr(x_1) -> 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_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_adx(x_1) -> adx(encArg(x_1))
   encode_n__adx(x_1) -> n__adx(encArg(x_1))
   encode_nats -> nats
   encode_zeros -> zeros
   encode_0 -> 0
   encode_n__zeros -> n__zeros
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(13) InfiniteLowerBoundProof (FINISHED)
The following loop proves infinite runtime complexity:

The rewrite sequence

activate(n__adx(cons(X1_0, n__zeros))) ->^+ cons(s(X1_0), n__incr(activate(n__adx(cons(0, n__zeros)))))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].

The pumping substitution is [ ].

The result substitution is [X1_0 / 0].




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

(14)
BOUNDS(INF, INF)