WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 195 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 37 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:

   remove(x', Cons(x, xs)) -> remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
   remove(x, Nil) -> Nil
   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
   minsort(Nil) -> Nil
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> minsort(xs)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   remove[Ite][True][Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
   remove[Ite][True][Ite](True, x', Cons(x, xs)) -> xs
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')

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:

   remove(x', Cons(x, xs)) -> remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
   remove(x, Nil) -> Nil
   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
   minsort(Nil) -> Nil
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> minsort(xs)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   remove[Ite][True][Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
   remove[Ite][True][Ite](True, x', Cons(x, xs)) -> xs
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')

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:

   remove(x', Cons(x, xs)) -> remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
   remove(x, Nil) -> Nil
   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
   minsort(Nil) -> Nil
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> minsort(xs)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   remove[Ite][True][Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
   remove[Ite][True][Ite](True, x', Cons(x, xs)) -> xs
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')

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

appmin(0, Cons(x, xs), xs') ->^+ appmin(0, xs, xs')

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 [ ].




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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:

   remove(x', Cons(x, xs)) -> remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
   remove(x, Nil) -> Nil
   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
   minsort(Nil) -> Nil
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> minsort(xs)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   remove[Ite][True][Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
   remove[Ite][True][Ite](True, x', Cons(x, xs)) -> xs
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')

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:

   remove(x', Cons(x, xs)) -> remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
   remove(x, Nil) -> Nil
   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
   minsort(Nil) -> Nil
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> minsort(xs)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   remove[Ite][True][Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
   remove[Ite][True][Ite](True, x', Cons(x, xs)) -> xs
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')

Rewrite Strategy: INNERMOST