/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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


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(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:

   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))
   a__hd(cons(X, Y)) -> mark(X)
   a__tl(cons(X, Y)) -> mark(Y)
   mark(nats) -> a__nats
   mark(adx(X)) -> a__adx(mark(X))
   mark(zeros) -> a__zeros
   mark(incr(X)) -> a__incr(mark(X))
   mark(hd(X)) -> a__hd(mark(X))
   mark(tl(X)) -> a__tl(mark(X))
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(0) -> 0
   mark(s(X)) -> s(X)
   a__nats -> nats
   a__adx(X) -> adx(X)
   a__zeros -> zeros
   a__incr(X) -> incr(X)
   a__hd(X) -> hd(X)
   a__tl(X) -> tl(X)

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(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:

   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))
   a__hd(cons(X, Y)) -> mark(X)
   a__tl(cons(X, Y)) -> mark(Y)
   mark(nats) -> a__nats
   mark(adx(X)) -> a__adx(mark(X))
   mark(zeros) -> a__zeros
   mark(incr(X)) -> a__incr(mark(X))
   mark(hd(X)) -> a__hd(mark(X))
   mark(tl(X)) -> a__tl(mark(X))
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(0) -> 0
   mark(s(X)) -> s(X)
   a__nats -> nats
   a__adx(X) -> adx(X)
   a__zeros -> zeros
   a__incr(X) -> incr(X)
   a__hd(X) -> hd(X)
   a__tl(X) -> tl(X)

S is empty.
Rewrite Strategy: FULL
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(3) 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 [ ].




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(4)
Complex Obligation (BEST)

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(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:

   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))
   a__hd(cons(X, Y)) -> mark(X)
   a__tl(cons(X, Y)) -> mark(Y)
   mark(nats) -> a__nats
   mark(adx(X)) -> a__adx(mark(X))
   mark(zeros) -> a__zeros
   mark(incr(X)) -> a__incr(mark(X))
   mark(hd(X)) -> a__hd(mark(X))
   mark(tl(X)) -> a__tl(mark(X))
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(0) -> 0
   mark(s(X)) -> s(X)
   a__nats -> nats
   a__adx(X) -> adx(X)
   a__zeros -> zeros
   a__incr(X) -> incr(X)
   a__hd(X) -> hd(X)
   a__tl(X) -> tl(X)

S is empty.
Rewrite Strategy: FULL
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(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(7)
BOUNDS(n^1, INF)

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(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:

   a__nats -> a__adx(a__zeros)
   a__zeros -> cons(0, zeros)
   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))
   a__hd(cons(X, Y)) -> mark(X)
   a__tl(cons(X, Y)) -> mark(Y)
   mark(nats) -> a__nats
   mark(adx(X)) -> a__adx(mark(X))
   mark(zeros) -> a__zeros
   mark(incr(X)) -> a__incr(mark(X))
   mark(hd(X)) -> a__hd(mark(X))
   mark(tl(X)) -> a__tl(mark(X))
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(0) -> 0
   mark(s(X)) -> s(X)
   a__nats -> nats
   a__adx(X) -> adx(X)
   a__zeros -> zeros
   a__incr(X) -> incr(X)
   a__hd(X) -> hd(X)
   a__tl(X) -> tl(X)

S is empty.
Rewrite Strategy: FULL