WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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), 211 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:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

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

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

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:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

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

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

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:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

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

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

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

subsets(Cons(x, xs)) ->^+ subsets[Ite][True][Let](Cons(x, xs), subsets(xs))

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

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

The result substitution is [ ].




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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:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

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

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

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:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

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

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

Rewrite Strategy: INNERMOST