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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   append(nil, x) -> cons(x, nil)
   append(cons(y, ys), x) -> cons(y, append(ys, x))
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
   if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs
   if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
   if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
   if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs)
   if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2)
   addList(xs, ys) -> addLists(xs, ys, nil)

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:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   append(nil, x) -> cons(x, nil)
   append(cons(y, ys), x) -> cons(y, append(ys, x))
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
   if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs
   if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
   if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
   if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs)
   if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2)
   addList(xs, ys) -> addLists(xs, ys, nil)

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

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 [ ].




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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:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   append(nil, x) -> cons(x, nil)
   append(cons(y, ys), x) -> cons(y, append(ys, x))
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
   if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs
   if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
   if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
   if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs)
   if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2)
   addList(xs, ys) -> addLists(xs, ys, nil)

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:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   append(nil, x) -> cons(x, nil)
   append(cons(y, ys), x) -> cons(y, append(ys, x))
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
   if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs
   if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
   if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
   if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs)
   if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2)
   addList(xs, ys) -> addLists(xs, ys, nil)

S is empty.
Rewrite Strategy: FULL