/export/starexec/sandbox/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, 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
        (9) DecreasingLoopProof [FINISHED, 25 ms]
        (10) BOUNDS(EXP, INF)


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS))
   from(X) -> cons(X, n__from(n__s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
   prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L)))
   app(X1, X2) -> n__app(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   nil -> n__nil
   zWadr(X1, X2) -> n__zWadr(X1, X2)
   prefix(X) -> n__prefix(X)
   activate(n__app(X1, X2)) -> app(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__nil) -> nil
   activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
   activate(n__prefix(X)) -> prefix(activate(X))
   activate(X) -> 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(EXP, INF).


The TRS R consists of the following rules:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS))
   from(X) -> cons(X, n__from(n__s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
   prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L)))
   app(X1, X2) -> n__app(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   nil -> n__nil
   zWadr(X1, X2) -> n__zWadr(X1, X2)
   prefix(X) -> n__prefix(X)
   activate(n__app(X1, X2)) -> app(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__nil) -> nil
   activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
   activate(n__prefix(X)) -> prefix(activate(X))
   activate(X) -> 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

activate(n__s(X)) ->^+ s(activate(X))

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

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


The TRS R consists of the following rules:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS))
   from(X) -> cons(X, n__from(n__s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
   prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L)))
   app(X1, X2) -> n__app(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   nil -> n__nil
   zWadr(X1, X2) -> n__zWadr(X1, X2)
   prefix(X) -> n__prefix(X)
   activate(n__app(X1, X2)) -> app(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__nil) -> nil
   activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
   activate(n__prefix(X)) -> prefix(activate(X))
   activate(X) -> 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(EXP, INF).


The TRS R consists of the following rules:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS))
   from(X) -> cons(X, n__from(n__s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
   prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L)))
   app(X1, X2) -> n__app(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   nil -> n__nil
   zWadr(X1, X2) -> n__zWadr(X1, X2)
   prefix(X) -> n__prefix(X)
   activate(n__app(X1, X2)) -> app(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__nil) -> nil
   activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
   activate(n__prefix(X)) -> prefix(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(9) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

activate(n__prefix(X)) ->^+ cons(nil, n__zWadr(activate(X), n__prefix(activate(X))))

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

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

The result substitution is [ ].



The rewrite sequence

activate(n__prefix(X)) ->^+ cons(nil, n__zWadr(activate(X), n__prefix(activate(X))))

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

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

The result substitution is [ ].




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(10)
BOUNDS(EXP, INF)