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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 382 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
    (22) CpxRNTS
    (23) FinalProof [FINISHED, 0 ms]
    (24) BOUNDS(1, n^1)
(25) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(26) TRS for Loop Detection
    (27) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (28) BEST
        (29) proven lower bound
            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
            (31) BOUNDS(n^1, INF)
        (32) TRS for Loop Detection


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


The TRS R consists of the following rules:

   rev(a) -> a
   rev(b) -> b
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(++(x, x)) -> rev(x)

S is empty.
Rewrite Strategy: FULL
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(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   rev(a) -> a
   rev(b) -> b
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(++(x, x)) -> rev(x)

S is empty.
Rewrite Strategy: INNERMOST
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(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   rev(a) -> a [1]
   rev(b) -> b [1]
   rev(++(x, y)) -> ++(rev(y), rev(x)) [1]
   rev(++(x, x)) -> rev(x) [1]

Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   rev(a) -> a [1]
   rev(b) -> b [1]
   rev(++(x, y)) -> ++(rev(y), rev(x)) [1]
   rev(++(x, x)) -> rev(x) [1]

The TRS has the following type information:
rev :: a:b:++ -> a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ -> a:b:++ -> a:b:++

Rewrite Strategy: INNERMOST
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(7) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   rev_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

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(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   rev(a) -> a [1]
   rev(b) -> b [1]
   rev(++(x, y)) -> ++(rev(y), rev(x)) [1]
   rev(++(x, x)) -> rev(x) [1]

The TRS has the following type information:
rev :: a:b:++ -> a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ -> a:b:++ -> a:b:++

Rewrite Strategy: INNERMOST
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(9) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   rev(a) -> a [1]
   rev(b) -> b [1]
   rev(++(x, y)) -> ++(rev(y), rev(x)) [1]
   rev(++(x, x)) -> rev(x) [1]

The TRS has the following type information:
rev :: a:b:++ -> a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ -> a:b:++ -> a:b:++

Rewrite Strategy: INNERMOST
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(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   a => 0
   b => 1

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(12)
Obligation:
Complexity RNTS consisting of the following rules:

   rev(z) -{ 1 }-> rev(x) :|: z = 1 + x + x, x >= 0
   rev(z) -{ 1 }-> 1 :|: z = 1
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + rev(y) + rev(x) :|: z = 1 + x + y, x >= 0, y >= 0


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(13) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(14)
Obligation:
Complexity RNTS consisting of the following rules:

   rev(z) -{ 1 }-> rev(x) :|: z = 1 + x + x, x >= 0
   rev(z) -{ 1 }-> 1 :|: z = 1
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + rev(y) + rev(x) :|: z = 1 + x + y, x >= 0, y >= 0


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(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { rev }

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(16)
Obligation:
Complexity RNTS consisting of the following rules:

   rev(z) -{ 1 }-> rev(x) :|: z = 1 + x + x, x >= 0
   rev(z) -{ 1 }-> 1 :|: z = 1
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + rev(y) + rev(x) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {rev}

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(17) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(18)
Obligation:
Complexity RNTS consisting of the following rules:

   rev(z) -{ 1 }-> rev(x) :|: z = 1 + x + x, x >= 0
   rev(z) -{ 1 }-> 1 :|: z = 1
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + rev(y) + rev(x) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {rev}

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(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: rev
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

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(20)
Obligation:
Complexity RNTS consisting of the following rules:

   rev(z) -{ 1 }-> rev(x) :|: z = 1 + x + x, x >= 0
   rev(z) -{ 1 }-> 1 :|: z = 1
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + rev(y) + rev(x) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {rev}
Previous analysis results are:
  rev: runtime: ?, size: O(n^1) [1 + z]

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(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: rev
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 2*z

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(22)
Obligation:
Complexity RNTS consisting of the following rules:

   rev(z) -{ 1 }-> rev(x) :|: z = 1 + x + x, x >= 0
   rev(z) -{ 1 }-> 1 :|: z = 1
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + rev(y) + rev(x) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  rev: runtime: O(n^1) [1 + 2*z], size: O(n^1) [1 + z]

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(23) FinalProof (FINISHED)
Computed overall runtime complexity
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(24)
BOUNDS(1, n^1)

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(25) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(26)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   rev(a) -> a
   rev(b) -> b
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(++(x, x)) -> rev(x)

S is empty.
Rewrite Strategy: FULL
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(27) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

rev(++(x, y)) ->^+ ++(rev(y), rev(x))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [y / ++(x, y)].

The result substitution is [ ].




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(28)
Complex Obligation (BEST)

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(29)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   rev(a) -> a
   rev(b) -> b
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(++(x, x)) -> rev(x)

S is empty.
Rewrite Strategy: FULL
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(30) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(31)
BOUNDS(n^1, INF)

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(32)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   rev(a) -> a
   rev(b) -> b
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(++(x, x)) -> rev(x)

S is empty.
Rewrite Strategy: FULL