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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(xs) -> sum2(xs, 0)
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
   ifsum(true, b, xs, y) -> y
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
   ifsum2(true, xs, y) -> sum2(tail(xs), y)
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
   isNil(nil) -> true
   isNil(cons(x, xs)) -> false
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   head(cons(x, xs)) -> x
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> c
   a -> d

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:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(xs) -> sum2(xs, 0)
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
   ifsum(true, b, xs, y) -> y
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
   ifsum2(true, xs, y) -> sum2(tail(xs), y)
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
   isNil(nil) -> true
   isNil(cons(x, xs)) -> false
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   head(cons(x, xs)) -> x
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> c
   a -> d

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

p(s(s(x))) ->^+ s(p(s(x)))

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

The pumping substitution is [x / s(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:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(xs) -> sum2(xs, 0)
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
   ifsum(true, b, xs, y) -> y
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
   ifsum2(true, xs, y) -> sum2(tail(xs), y)
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
   isNil(nil) -> true
   isNil(cons(x, xs)) -> false
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   head(cons(x, xs)) -> x
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> c
   a -> d

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:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(xs) -> sum2(xs, 0)
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
   ifsum(true, b, xs, y) -> y
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
   ifsum2(true, xs, y) -> sum2(tail(xs), y)
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
   isNil(nil) -> true
   isNil(cons(x, xs)) -> false
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   head(cons(x, xs)) -> x
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> c
   a -> d

S is empty.
Rewrite Strategy: FULL