/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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:

   isEmpty(empty) -> true
   isEmpty(node(l, x, r)) -> false
   left(empty) -> empty
   left(node(l, x, r)) -> l
   right(empty) -> empty
   right(node(l, x, r)) -> r
   elem(node(l, x, r)) -> x
   append(nil, x) -> cons(x, nil)
   append(cons(y, ys), x) -> cons(y, append(ys, x))
   listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n))
   if(true, b, n, m, xs, ys) -> xs
   if(false, false, n, m, xs, ys) -> listify(m, xs)
   if(false, true, n, m, xs, ys) -> listify(n, ys)
   toList(n) -> listify(n, nil)

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:

   isEmpty(empty) -> true
   isEmpty(node(l, x, r)) -> false
   left(empty) -> empty
   left(node(l, x, r)) -> l
   right(empty) -> empty
   right(node(l, x, r)) -> r
   elem(node(l, x, r)) -> x
   append(nil, x) -> cons(x, nil)
   append(cons(y, ys), x) -> cons(y, append(ys, x))
   listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n))
   if(true, b, n, m, xs, ys) -> xs
   if(false, false, n, m, xs, ys) -> listify(m, xs)
   if(false, true, n, m, xs, ys) -> listify(n, ys)
   toList(n) -> listify(n, nil)

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

append(cons(y, ys), x) ->^+ cons(y, append(ys, x))

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

The pumping substitution is [ys / cons(y, ys)].

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:

   isEmpty(empty) -> true
   isEmpty(node(l, x, r)) -> false
   left(empty) -> empty
   left(node(l, x, r)) -> l
   right(empty) -> empty
   right(node(l, x, r)) -> r
   elem(node(l, x, r)) -> x
   append(nil, x) -> cons(x, nil)
   append(cons(y, ys), x) -> cons(y, append(ys, x))
   listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n))
   if(true, b, n, m, xs, ys) -> xs
   if(false, false, n, m, xs, ys) -> listify(m, xs)
   if(false, true, n, m, xs, ys) -> listify(n, ys)
   toList(n) -> listify(n, nil)

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:

   isEmpty(empty) -> true
   isEmpty(node(l, x, r)) -> false
   left(empty) -> empty
   left(node(l, x, r)) -> l
   right(empty) -> empty
   right(node(l, x, r)) -> r
   elem(node(l, x, r)) -> x
   append(nil, x) -> cons(x, nil)
   append(cons(y, ys), x) -> cons(y, append(ys, x))
   listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n))
   if(true, b, n, m, xs, ys) -> xs
   if(false, false, n, m, xs, ys) -> listify(m, xs)
   if(false, true, n, m, xs, ys) -> listify(n, ys)
   toList(n) -> listify(n, nil)

S is empty.
Rewrite Strategy: FULL