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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 588 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 233 ms]
    (20) CpxRNTS
    (21) FinalProof [FINISHED, 0 ms]
    (22) BOUNDS(1, n^1)
(23) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(24) TRS for Loop Detection
    (25) DecreasingLoopProof [LOWER BOUND(ID), 17 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) TRS for Loop Detection


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


The TRS R consists of the following rules:

   f(empty, l) -> l
   f(cons(x, k), l) -> g(k, l, cons(x, k))
   g(a, b, c) -> f(a, cons(b, c))

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(2)
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:

   f(empty, l) -> l [1]
   f(cons(x, k), l) -> g(k, l, cons(x, k)) [1]
   g(a, b, c) -> f(a, cons(b, c)) [1]

Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(empty, l) -> l [1]
   f(cons(x, k), l) -> g(k, l, cons(x, k)) [1]
   g(a, b, c) -> f(a, cons(b, c)) [1]

The TRS has the following type information:
f :: empty:cons -> empty:cons -> empty:cons
empty :: empty:cons
cons :: empty:cons -> empty:cons -> empty:cons
g :: empty:cons -> empty:cons -> empty:cons -> empty:cons

Rewrite Strategy: INNERMOST
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(5) 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:

   f_2
   g_3

(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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(6)
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:

   f(empty, l) -> l [1]
   f(cons(x, k), l) -> g(k, l, cons(x, k)) [1]
   g(a, b, c) -> f(a, cons(b, c)) [1]

The TRS has the following type information:
f :: empty:cons -> empty:cons -> empty:cons
empty :: empty:cons
cons :: empty:cons -> empty:cons -> empty:cons
g :: empty:cons -> empty:cons -> empty:cons -> empty:cons

Rewrite Strategy: INNERMOST
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(7) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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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:

   f(empty, l) -> l [1]
   f(cons(x, k), l) -> g(k, l, cons(x, k)) [1]
   g(a, b, c) -> f(a, cons(b, c)) [1]

The TRS has the following type information:
f :: empty:cons -> empty:cons -> empty:cons
empty :: empty:cons
cons :: empty:cons -> empty:cons -> empty:cons
g :: empty:cons -> empty:cons -> empty:cons -> empty:cons

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

   empty => 0

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(10)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> l :|: z' = l, l >= 0, z = 0
   f(z, z') -{ 1 }-> g(k, l, 1 + x + k) :|: z' = l, x >= 0, l >= 0, k >= 0, z = 1 + x + k
   g(z, z', z'') -{ 1 }-> f(a, 1 + b + c) :|: z = a, b >= 0, a >= 0, c >= 0, z' = b, z'' = c


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(11) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(12)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k
   g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0


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

   { f, g }

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(14)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k
   g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0

Function symbols to be analyzed: {f,g}

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(15) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(16)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k
   g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0

Function symbols to be analyzed: {f,g}

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(17) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z + z^2 + z'

Computed SIZE bound using KoAT for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + z + z^2 + z' + z''

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(18)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k
   g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0

Function symbols to be analyzed: {f,g}
Previous analysis results are:
  f: runtime: ?, size: O(n^2) [z + z^2 + z']
  g: runtime: ?, size: O(n^2) [1 + z + z^2 + z' + z'']

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(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 2*z

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 2*z

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

   f(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z, z') -{ 1 }-> g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k
   g(z, z', z'') -{ 1 }-> f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  f: runtime: O(n^1) [3 + 2*z], size: O(n^2) [z + z^2 + z']
  g: runtime: O(n^1) [4 + 2*z], size: O(n^2) [1 + z + z^2 + z' + z'']

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(21) FinalProof (FINISHED)
Computed overall runtime complexity
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(22)
BOUNDS(1, n^1)

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(23) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(24)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(empty, l) -> l
   f(cons(x, k), l) -> g(k, l, cons(x, k))
   g(a, b, c) -> f(a, cons(b, c))

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

The rewrite sequence

g(cons(x1_0, k2_0), b, c) ->^+ g(k2_0, cons(b, c), cons(x1_0, k2_0))

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

The pumping substitution is [k2_0 / cons(x1_0, k2_0)].

The result substitution is [b / cons(b, c), c / cons(x1_0, k2_0)].




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(26)
Complex Obligation (BEST)

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(27)
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(n^1, n^1).


The TRS R consists of the following rules:

   f(empty, l) -> l
   f(cons(x, k), l) -> g(k, l, cons(x, k))
   g(a, b, c) -> f(a, cons(b, c))

S is empty.
Rewrite Strategy: INNERMOST
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(28) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(29)
BOUNDS(n^1, INF)

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(30)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(empty, l) -> l
   f(cons(x, k), l) -> g(k, l, cons(x, k))
   g(a, b, c) -> f(a, cons(b, c))

S is empty.
Rewrite Strategy: INNERMOST