/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 106 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:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   and(tt, X) -> activate(X)
   isList(V) -> isNeList(activate(V))
   isList(n__nil) -> tt
   isList(n____(V1, V2)) -> and(isList(activate(V1)), n__isList(activate(V2)))
   isNeList(V) -> isQid(activate(V))
   isNeList(n____(V1, V2)) -> and(isList(activate(V1)), n__isNeList(activate(V2)))
   isNeList(n____(V1, V2)) -> and(isNeList(activate(V1)), n__isList(activate(V2)))
   isNePal(V) -> isQid(activate(V))
   isNePal(n____(I, __(P, I))) -> and(isQid(activate(I)), n__isPal(activate(P)))
   isPal(V) -> isNePal(activate(V))
   isPal(n__nil) -> tt
   isQid(n__a) -> tt
   isQid(n__e) -> tt
   isQid(n__i) -> tt
   isQid(n__o) -> tt
   isQid(n__u) -> tt
   nil -> n__nil
   __(X1, X2) -> n____(X1, X2)
   isList(X) -> n__isList(X)
   isNeList(X) -> n__isNeList(X)
   isPal(X) -> n__isPal(X)
   a -> n__a
   e -> n__e
   i -> n__i
   o -> n__o
   u -> n__u
   activate(n__nil) -> nil
   activate(n____(X1, X2)) -> __(X1, X2)
   activate(n__isList(X)) -> isList(X)
   activate(n__isNeList(X)) -> isNeList(X)
   activate(n__isPal(X)) -> isPal(X)
   activate(n__a) -> a
   activate(n__e) -> e
   activate(n__i) -> i
   activate(n__o) -> o
   activate(n__u) -> u
   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.
----------------------------------------

(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:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   and(tt, X) -> activate(X)
   isList(V) -> isNeList(activate(V))
   isList(n__nil) -> tt
   isList(n____(V1, V2)) -> and(isList(activate(V1)), n__isList(activate(V2)))
   isNeList(V) -> isQid(activate(V))
   isNeList(n____(V1, V2)) -> and(isList(activate(V1)), n__isNeList(activate(V2)))
   isNeList(n____(V1, V2)) -> and(isNeList(activate(V1)), n__isList(activate(V2)))
   isNePal(V) -> isQid(activate(V))
   isNePal(n____(I, __(P, I))) -> and(isQid(activate(I)), n__isPal(activate(P)))
   isPal(V) -> isNePal(activate(V))
   isPal(n__nil) -> tt
   isQid(n__a) -> tt
   isQid(n__e) -> tt
   isQid(n__i) -> tt
   isQid(n__o) -> tt
   isQid(n__u) -> tt
   nil -> n__nil
   __(X1, X2) -> n____(X1, X2)
   isList(X) -> n__isList(X)
   isNeList(X) -> n__isNeList(X)
   isPal(X) -> n__isPal(X)
   a -> n__a
   e -> n__e
   i -> n__i
   o -> n__o
   u -> n__u
   activate(n__nil) -> nil
   activate(n____(X1, X2)) -> __(X1, X2)
   activate(n__isList(X)) -> isList(X)
   activate(n__isNeList(X)) -> isNeList(X)
   activate(n__isPal(X)) -> isPal(X)
   activate(n__a) -> a
   activate(n__e) -> e
   activate(n__i) -> i
   activate(n__o) -> o
   activate(n__u) -> u
   activate(X) -> 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

isList(n____(n__isList(X1_0), V2)) ->^+ and(isList(isList(X1_0)), n__isList(activate(V2)))

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

The pumping substitution is [X1_0 / n____(n__isList(X1_0), V2)].

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:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   and(tt, X) -> activate(X)
   isList(V) -> isNeList(activate(V))
   isList(n__nil) -> tt
   isList(n____(V1, V2)) -> and(isList(activate(V1)), n__isList(activate(V2)))
   isNeList(V) -> isQid(activate(V))
   isNeList(n____(V1, V2)) -> and(isList(activate(V1)), n__isNeList(activate(V2)))
   isNeList(n____(V1, V2)) -> and(isNeList(activate(V1)), n__isList(activate(V2)))
   isNePal(V) -> isQid(activate(V))
   isNePal(n____(I, __(P, I))) -> and(isQid(activate(I)), n__isPal(activate(P)))
   isPal(V) -> isNePal(activate(V))
   isPal(n__nil) -> tt
   isQid(n__a) -> tt
   isQid(n__e) -> tt
   isQid(n__i) -> tt
   isQid(n__o) -> tt
   isQid(n__u) -> tt
   nil -> n__nil
   __(X1, X2) -> n____(X1, X2)
   isList(X) -> n__isList(X)
   isNeList(X) -> n__isNeList(X)
   isPal(X) -> n__isPal(X)
   a -> n__a
   e -> n__e
   i -> n__i
   o -> n__o
   u -> n__u
   activate(n__nil) -> nil
   activate(n____(X1, X2)) -> __(X1, X2)
   activate(n__isList(X)) -> isList(X)
   activate(n__isNeList(X)) -> isNeList(X)
   activate(n__isPal(X)) -> isPal(X)
   activate(n__a) -> a
   activate(n__e) -> e
   activate(n__i) -> i
   activate(n__o) -> o
   activate(n__u) -> u
   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(n^1, INF).


The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   and(tt, X) -> activate(X)
   isList(V) -> isNeList(activate(V))
   isList(n__nil) -> tt
   isList(n____(V1, V2)) -> and(isList(activate(V1)), n__isList(activate(V2)))
   isNeList(V) -> isQid(activate(V))
   isNeList(n____(V1, V2)) -> and(isList(activate(V1)), n__isNeList(activate(V2)))
   isNeList(n____(V1, V2)) -> and(isNeList(activate(V1)), n__isList(activate(V2)))
   isNePal(V) -> isQid(activate(V))
   isNePal(n____(I, __(P, I))) -> and(isQid(activate(I)), n__isPal(activate(P)))
   isPal(V) -> isNePal(activate(V))
   isPal(n__nil) -> tt
   isQid(n__a) -> tt
   isQid(n__e) -> tt
   isQid(n__i) -> tt
   isQid(n__o) -> tt
   isQid(n__u) -> tt
   nil -> n__nil
   __(X1, X2) -> n____(X1, X2)
   isList(X) -> n__isList(X)
   isNeList(X) -> n__isNeList(X)
   isPal(X) -> n__isPal(X)
   a -> n__a
   e -> n__e
   i -> n__i
   o -> n__o
   u -> n__u
   activate(n__nil) -> nil
   activate(n____(X1, X2)) -> __(X1, X2)
   activate(n__isList(X)) -> isList(X)
   activate(n__isNeList(X)) -> isNeList(X)
   activate(n__isPal(X)) -> isPal(X)
   activate(n__a) -> a
   activate(n__e) -> e
   activate(n__i) -> i
   activate(n__o) -> o
   activate(n__u) -> u
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL