/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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:

   active(primes) -> mark(sieve(from(s(s(0)))))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(head(cons(X, Y))) -> mark(X)
   active(tail(cons(X, Y))) -> mark(Y)
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
   active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y))))
   active(sieve(X)) -> sieve(active(X))
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(filter(X1, X2)) -> filter(active(X1), X2)
   active(filter(X1, X2)) -> filter(X1, active(X2))
   active(divides(X1, X2)) -> divides(active(X1), X2)
   active(divides(X1, X2)) -> divides(X1, active(X2))
   sieve(mark(X)) -> mark(sieve(X))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   filter(mark(X1), X2) -> mark(filter(X1, X2))
   filter(X1, mark(X2)) -> mark(filter(X1, X2))
   divides(mark(X1), X2) -> mark(divides(X1, X2))
   divides(X1, mark(X2)) -> mark(divides(X1, X2))
   proper(primes) -> ok(primes)
   proper(sieve(X)) -> sieve(proper(X))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(filter(X1, X2)) -> filter(proper(X1), proper(X2))
   proper(divides(X1, X2)) -> divides(proper(X1), proper(X2))
   sieve(ok(X)) -> ok(sieve(X))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   filter(ok(X1), ok(X2)) -> ok(filter(X1, X2))
   divides(ok(X1), ok(X2)) -> ok(divides(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(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:

   active(primes) -> mark(sieve(from(s(s(0)))))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(head(cons(X, Y))) -> mark(X)
   active(tail(cons(X, Y))) -> mark(Y)
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
   active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y))))
   active(sieve(X)) -> sieve(active(X))
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(filter(X1, X2)) -> filter(active(X1), X2)
   active(filter(X1, X2)) -> filter(X1, active(X2))
   active(divides(X1, X2)) -> divides(active(X1), X2)
   active(divides(X1, X2)) -> divides(X1, active(X2))
   sieve(mark(X)) -> mark(sieve(X))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   filter(mark(X1), X2) -> mark(filter(X1, X2))
   filter(X1, mark(X2)) -> mark(filter(X1, X2))
   divides(mark(X1), X2) -> mark(divides(X1, X2))
   divides(X1, mark(X2)) -> mark(divides(X1, X2))
   proper(primes) -> ok(primes)
   proper(sieve(X)) -> sieve(proper(X))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(filter(X1, X2)) -> filter(proper(X1), proper(X2))
   proper(divides(X1, X2)) -> divides(proper(X1), proper(X2))
   sieve(ok(X)) -> ok(sieve(X))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   filter(ok(X1), ok(X2)) -> ok(filter(X1, X2))
   divides(ok(X1), ok(X2)) -> ok(divides(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(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

tail(ok(X)) ->^+ ok(tail(X))

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

The pumping substitution is [X / ok(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:

   active(primes) -> mark(sieve(from(s(s(0)))))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(head(cons(X, Y))) -> mark(X)
   active(tail(cons(X, Y))) -> mark(Y)
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
   active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y))))
   active(sieve(X)) -> sieve(active(X))
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(filter(X1, X2)) -> filter(active(X1), X2)
   active(filter(X1, X2)) -> filter(X1, active(X2))
   active(divides(X1, X2)) -> divides(active(X1), X2)
   active(divides(X1, X2)) -> divides(X1, active(X2))
   sieve(mark(X)) -> mark(sieve(X))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   filter(mark(X1), X2) -> mark(filter(X1, X2))
   filter(X1, mark(X2)) -> mark(filter(X1, X2))
   divides(mark(X1), X2) -> mark(divides(X1, X2))
   divides(X1, mark(X2)) -> mark(divides(X1, X2))
   proper(primes) -> ok(primes)
   proper(sieve(X)) -> sieve(proper(X))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(filter(X1, X2)) -> filter(proper(X1), proper(X2))
   proper(divides(X1, X2)) -> divides(proper(X1), proper(X2))
   sieve(ok(X)) -> ok(sieve(X))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   filter(ok(X1), ok(X2)) -> ok(filter(X1, X2))
   divides(ok(X1), ok(X2)) -> ok(divides(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(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:

   active(primes) -> mark(sieve(from(s(s(0)))))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(head(cons(X, Y))) -> mark(X)
   active(tail(cons(X, Y))) -> mark(Y)
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
   active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y))))
   active(sieve(X)) -> sieve(active(X))
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(filter(X1, X2)) -> filter(active(X1), X2)
   active(filter(X1, X2)) -> filter(X1, active(X2))
   active(divides(X1, X2)) -> divides(active(X1), X2)
   active(divides(X1, X2)) -> divides(X1, active(X2))
   sieve(mark(X)) -> mark(sieve(X))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   filter(mark(X1), X2) -> mark(filter(X1, X2))
   filter(X1, mark(X2)) -> mark(filter(X1, X2))
   divides(mark(X1), X2) -> mark(divides(X1, X2))
   divides(X1, mark(X2)) -> mark(divides(X1, X2))
   proper(primes) -> ok(primes)
   proper(sieve(X)) -> sieve(proper(X))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(filter(X1, X2)) -> filter(proper(X1), proper(X2))
   proper(divides(X1, X2)) -> divides(proper(X1), proper(X2))
   sieve(ok(X)) -> ok(sieve(X))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   filter(ok(X1), ok(X2)) -> ok(filter(X1, X2))
   divides(ok(X1), ok(X2)) -> ok(divides(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL