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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 169 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(6) BEST
    (7) proven lower bound
        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
        (9) BOUNDS(n^1, INF)
    (10) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

The (relative) TRS S consists of the following rules:

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

The (relative) TRS S consists of the following rules:

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Rewrite Strategy: INNERMOST
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(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(4)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

The (relative) TRS S consists of the following rules:

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

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

The rewrite sequence

splitmerge(Cons(x, xs), xs1, xs2) ->^+ splitmerge(xs, Cons(x, xs2), xs1)

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

The pumping substitution is [xs / Cons(x, xs)].

The result substitution is [xs1 / Cons(x, xs2), xs2 / xs1].




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(6)
Complex Obligation (BEST)

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(7)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

The (relative) TRS S consists of the following rules:

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Rewrite Strategy: INNERMOST
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(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(9)
BOUNDS(n^1, INF)

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(10)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

The (relative) TRS S consists of the following rules:

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Rewrite Strategy: INNERMOST