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


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


WORST_CASE(Omega(n^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(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


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

(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__dbl(0) -> 0
   a__dbl(s(X)) -> s(s(dbl(X)))
   a__dbls(nil) -> nil
   a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   a__sel(0, cons(X, Y)) -> mark(X)
   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
   a__indx(nil, X) -> nil
   a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   a__from(X) -> cons(X, from(s(X)))
   a__dbl1(0) -> 01
   a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X))))
   a__sel1(0, cons(X, Y)) -> mark(X)
   a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z))
   a__quote(0) -> 01
   a__quote(s(X)) -> s1(a__quote(mark(X)))
   a__quote(dbl(X)) -> a__dbl1(mark(X))
   a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y))
   mark(dbl(X)) -> a__dbl(mark(X))
   mark(dbls(X)) -> a__dbls(mark(X))
   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
   mark(indx(X1, X2)) -> a__indx(mark(X1), X2)
   mark(from(X)) -> a__from(X)
   mark(dbl1(X)) -> a__dbl1(mark(X))
   mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2))
   mark(quote(X)) -> a__quote(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(01) -> 01
   mark(s1(X)) -> s1(mark(X))
   a__dbl(X) -> dbl(X)
   a__dbls(X) -> dbls(X)
   a__sel(X1, X2) -> sel(X1, X2)
   a__indx(X1, X2) -> indx(X1, X2)
   a__from(X) -> from(X)
   a__dbl1(X) -> dbl1(X)
   a__sel1(X1, X2) -> sel1(X1, X2)
   a__quote(X) -> quote(X)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(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:

   a__dbl(0) -> 0
   a__dbl(s(X)) -> s(s(dbl(X)))
   a__dbls(nil) -> nil
   a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   a__sel(0, cons(X, Y)) -> mark(X)
   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
   a__indx(nil, X) -> nil
   a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   a__from(X) -> cons(X, from(s(X)))
   a__dbl1(0) -> 01
   a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X))))
   a__sel1(0, cons(X, Y)) -> mark(X)
   a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z))
   a__quote(0) -> 01
   a__quote(s(X)) -> s1(a__quote(mark(X)))
   a__quote(dbl(X)) -> a__dbl1(mark(X))
   a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y))
   mark(dbl(X)) -> a__dbl(mark(X))
   mark(dbls(X)) -> a__dbls(mark(X))
   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
   mark(indx(X1, X2)) -> a__indx(mark(X1), X2)
   mark(from(X)) -> a__from(X)
   mark(dbl1(X)) -> a__dbl1(mark(X))
   mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2))
   mark(quote(X)) -> a__quote(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(01) -> 01
   mark(s1(X)) -> s1(mark(X))
   a__dbl(X) -> dbl(X)
   a__dbls(X) -> dbls(X)
   a__sel(X1, X2) -> sel(X1, X2)
   a__indx(X1, X2) -> indx(X1, X2)
   a__from(X) -> from(X)
   a__dbl1(X) -> dbl1(X)
   a__sel1(X1, X2) -> sel1(X1, X2)
   a__quote(X) -> quote(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

mark(indx(X1, X2)) ->^+ a__indx(mark(X1), X2)

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

The pumping substitution is [X1 / indx(X1, X2)].

The result substitution is [ ].




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

(4)
Complex Obligation (BEST)

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

(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__dbl(0) -> 0
   a__dbl(s(X)) -> s(s(dbl(X)))
   a__dbls(nil) -> nil
   a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   a__sel(0, cons(X, Y)) -> mark(X)
   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
   a__indx(nil, X) -> nil
   a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   a__from(X) -> cons(X, from(s(X)))
   a__dbl1(0) -> 01
   a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X))))
   a__sel1(0, cons(X, Y)) -> mark(X)
   a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z))
   a__quote(0) -> 01
   a__quote(s(X)) -> s1(a__quote(mark(X)))
   a__quote(dbl(X)) -> a__dbl1(mark(X))
   a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y))
   mark(dbl(X)) -> a__dbl(mark(X))
   mark(dbls(X)) -> a__dbls(mark(X))
   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
   mark(indx(X1, X2)) -> a__indx(mark(X1), X2)
   mark(from(X)) -> a__from(X)
   mark(dbl1(X)) -> a__dbl1(mark(X))
   mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2))
   mark(quote(X)) -> a__quote(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(01) -> 01
   mark(s1(X)) -> s1(mark(X))
   a__dbl(X) -> dbl(X)
   a__dbls(X) -> dbls(X)
   a__sel(X1, X2) -> sel(X1, X2)
   a__indx(X1, X2) -> indx(X1, X2)
   a__from(X) -> from(X)
   a__dbl1(X) -> dbl1(X)
   a__sel1(X1, X2) -> sel1(X1, X2)
   a__quote(X) -> quote(X)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(7)
BOUNDS(n^1, INF)

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

(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__dbl(0) -> 0
   a__dbl(s(X)) -> s(s(dbl(X)))
   a__dbls(nil) -> nil
   a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   a__sel(0, cons(X, Y)) -> mark(X)
   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
   a__indx(nil, X) -> nil
   a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   a__from(X) -> cons(X, from(s(X)))
   a__dbl1(0) -> 01
   a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X))))
   a__sel1(0, cons(X, Y)) -> mark(X)
   a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z))
   a__quote(0) -> 01
   a__quote(s(X)) -> s1(a__quote(mark(X)))
   a__quote(dbl(X)) -> a__dbl1(mark(X))
   a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y))
   mark(dbl(X)) -> a__dbl(mark(X))
   mark(dbls(X)) -> a__dbls(mark(X))
   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
   mark(indx(X1, X2)) -> a__indx(mark(X1), X2)
   mark(from(X)) -> a__from(X)
   mark(dbl1(X)) -> a__dbl1(mark(X))
   mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2))
   mark(quote(X)) -> a__quote(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(01) -> 01
   mark(s1(X)) -> s1(mark(X))
   a__dbl(X) -> dbl(X)
   a__dbls(X) -> dbls(X)
   a__sel(X1, X2) -> sel(X1, X2)
   a__indx(X1, X2) -> indx(X1, X2)
   a__from(X) -> from(X)
   a__dbl1(X) -> dbl1(X)
   a__sel1(X1, X2) -> sel1(X1, X2)
   a__quote(X) -> quote(X)

S is empty.
Rewrite Strategy: FULL