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


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, 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
        (9) DecreasingLoopProof [FINISHED, 595 ms]
        (10) BOUNDS(EXP, INF)


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   select(x', revprefix, Cons(x, xs)) -> mapconsapp(x', permute(revapp(revprefix, Cons(x, xs))), select(x, Cons(x', revprefix), xs))
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
   permute(Cons(x, xs)) -> select(x, Nil, xs)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   select(x, revprefix, Nil) -> mapconsapp(x, permute(revapp(revprefix, Nil)), Nil)
   revapp(Nil, rest) -> rest
   permute(Nil) -> Cons(Nil, Nil)
   mapconsapp(x, Nil, rest) -> rest
   goal(xs) -> permute(xs)

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   select(x', revprefix, Cons(x, xs)) -> mapconsapp(x', permute(revapp(revprefix, Cons(x, xs))), select(x, Cons(x', revprefix), xs))
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
   permute(Cons(x, xs)) -> select(x, Nil, xs)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   select(x, revprefix, Nil) -> mapconsapp(x, permute(revapp(revprefix, Nil)), Nil)
   revapp(Nil, rest) -> rest
   permute(Nil) -> Cons(Nil, Nil)
   mapconsapp(x, Nil, rest) -> rest
   goal(xs) -> permute(xs)

S is empty.
Rewrite Strategy: INNERMOST
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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

revapp(Cons(x, xs), rest) ->^+ revapp(xs, Cons(x, rest))

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 [rest / Cons(x, rest)].




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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   select(x', revprefix, Cons(x, xs)) -> mapconsapp(x', permute(revapp(revprefix, Cons(x, xs))), select(x, Cons(x', revprefix), xs))
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
   permute(Cons(x, xs)) -> select(x, Nil, xs)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   select(x, revprefix, Nil) -> mapconsapp(x, permute(revapp(revprefix, Nil)), Nil)
   revapp(Nil, rest) -> rest
   permute(Nil) -> Cons(Nil, Nil)
   mapconsapp(x, Nil, rest) -> rest
   goal(xs) -> permute(xs)

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   select(x', revprefix, Cons(x, xs)) -> mapconsapp(x', permute(revapp(revprefix, Cons(x, xs))), select(x, Cons(x', revprefix), xs))
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
   permute(Cons(x, xs)) -> select(x, Nil, xs)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   select(x, revprefix, Nil) -> mapconsapp(x, permute(revapp(revprefix, Nil)), Nil)
   revapp(Nil, rest) -> rest
   permute(Nil) -> Cons(Nil, Nil)
   mapconsapp(x, Nil, rest) -> rest
   goal(xs) -> permute(xs)

S is empty.
Rewrite Strategy: INNERMOST
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(9) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

permute(Cons(x, Cons(x3_0, Cons(x3_1, xs4_1)))) ->^+ mapconsapp(x, permute(Cons(x3_0, Cons(x3_1, xs4_1))), mapconsapp(x3_0, permute(Cons(x, Cons(x3_1, xs4_1))), select(x3_1, Cons(x3_0, Cons(x, Nil)), xs4_1)))

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

The pumping substitution is [xs4_1 / Cons(x3_1, xs4_1)].

The result substitution is [x / x3_0, x3_0 / x3_1].



The rewrite sequence

permute(Cons(x, Cons(x3_0, Cons(x3_1, xs4_1)))) ->^+ mapconsapp(x, permute(Cons(x3_0, Cons(x3_1, xs4_1))), mapconsapp(x3_0, permute(Cons(x, Cons(x3_1, xs4_1))), select(x3_1, Cons(x3_0, Cons(x, Nil)), xs4_1)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].

The pumping substitution is [xs4_1 / Cons(x3_1, xs4_1)].

The result substitution is [x3_0 / x3_1].




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(10)
BOUNDS(EXP, INF)