/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(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (innermost) 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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__fst(0, Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__from(X) -> cons(mark(X), from(s(X)))
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   a__len(nil) -> 0
   a__len(cons(X, Z)) -> s(len(Z))
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(len(X)) -> a__len(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__fst(X1, X2) -> fst(X1, X2)
   a__from(X) -> from(X)
   a__add(X1, X2) -> add(X1, X2)
   a__len(X) -> len(X)

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__fst(0, Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__from(X) -> cons(mark(X), from(s(X)))
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   a__len(nil) -> 0
   a__len(cons(X, Z)) -> s(len(Z))
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(len(X)) -> a__len(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__fst(X1, X2) -> fst(X1, X2)
   a__from(X) -> from(X)
   a__add(X1, X2) -> add(X1, X2)
   a__len(X) -> len(X)

S is empty.
Rewrite Strategy: INNERMOST
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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(len(X)) ->^+ a__len(mark(X))

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

The pumping substitution is [X / len(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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__fst(0, Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__from(X) -> cons(mark(X), from(s(X)))
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   a__len(nil) -> 0
   a__len(cons(X, Z)) -> s(len(Z))
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(len(X)) -> a__len(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__fst(X1, X2) -> fst(X1, X2)
   a__from(X) -> from(X)
   a__add(X1, X2) -> add(X1, X2)
   a__len(X) -> len(X)

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__fst(0, Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__from(X) -> cons(mark(X), from(s(X)))
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   a__len(nil) -> 0
   a__len(cons(X, Z)) -> s(len(Z))
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(len(X)) -> a__len(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__fst(X1, X2) -> fst(X1, X2)
   a__from(X) -> from(X)
   a__add(X1, X2) -> add(X1, X2)
   a__len(X) -> len(X)

S is empty.
Rewrite Strategy: INNERMOST