/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (innermost) 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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
   if(true, x, y, z, u) -> z
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
   if2(true, x, y, z, u) -> z
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> ifTimes(ge(0, x), x, y)
   ifTimes(true, x, y) -> 0
   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
   p(s(x)) -> x
   p(0) -> s(s(0))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   or(true, y) -> true
   or(false, y) -> y
   divisible(0, s(y)) -> true
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
   div(x, y, 0) -> divisible(x, y)
   div(0, y, s(z)) -> false
   div(s(x), y, s(z)) -> div(x, y, z)
   a -> b
   a -> c

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

(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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
   if(true, x, y, z, u) -> z
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
   if2(true, x, y, z, u) -> z
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> ifTimes(ge(0, x), x, y)
   ifTimes(true, x, y) -> 0
   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
   p(s(x)) -> x
   p(0) -> s(s(0))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   or(true, y) -> true
   or(false, y) -> y
   divisible(0, s(y)) -> true
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
   div(x, y, 0) -> divisible(x, y)
   div(0, y, s(z)) -> false
   div(s(x), y, s(z)) -> div(x, y, z)
   a -> b
   a -> c

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

(3) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

div(s(x), y, s(z)) ->^+ div(x, y, z)

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

The pumping substitution is [x / s(x), z / s(z)].

The result substitution is [ ].




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

(4)
Complex Obligation (BEST)

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

(5)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
   if(true, x, y, z, u) -> z
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
   if2(true, x, y, z, u) -> z
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> ifTimes(ge(0, x), x, y)
   ifTimes(true, x, y) -> 0
   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
   p(s(x)) -> x
   p(0) -> s(s(0))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   or(true, y) -> true
   or(false, y) -> y
   divisible(0, s(y)) -> true
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
   div(x, y, 0) -> divisible(x, y)
   div(0, y, s(z)) -> false
   div(s(x), y, s(z)) -> div(x, y, z)
   a -> b
   a -> c

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

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

(7)
BOUNDS(n^1, INF)

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

(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
   if(true, x, y, z, u) -> z
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
   if2(true, x, y, z, u) -> z
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> ifTimes(ge(0, x), x, y)
   ifTimes(true, x, y) -> 0
   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
   p(s(x)) -> x
   p(0) -> s(s(0))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   or(true, y) -> true
   or(false, y) -> y
   divisible(0, s(y)) -> true
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
   div(x, y, 0) -> divisible(x, y)
   div(0, y, s(z)) -> false
   div(s(x), y, s(z)) -> div(x, y, z)
   a -> b
   a -> c

S is empty.
Rewrite Strategy: INNERMOST