WORST_CASE(?,O(n^1))
proof of input_WKn5lRvwua.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
(3) CpxWeightedTrsRenamingProof [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), 238 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 87 ms]
(22) CpxRNTS
(23) FinalProof [FINISHED, 0 ms]
(24) BOUNDS(1, n^1)


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


The TRS R consists of the following rules:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RelTrsToWeightedTrsProof (UPPER BOUND(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:

   *(x, +(y, z)) -> +(*(x, y), *(x, z)) [1]

Rewrite Strategy: INNERMOST
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(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

* => times

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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:

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [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:

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]

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

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:

   times_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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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:

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]

The TRS has the following type information:
times :: a -> + -> +
+ :: + -> + -> +
const :: +
const1 :: a

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:

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]

The TRS has the following type information:
times :: a -> + -> +
+ :: + -> + -> +
const :: +
const1 :: a

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:

   const => 0
   const1 => 0

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

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


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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:

   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z


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

   { times }

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

   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {times}

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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:

   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {times}

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(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {times}
Previous analysis results are:
  times: runtime: ?, size: O(1) [0]

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(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: 
Previous analysis results are:
  times: runtime: O(n^1) [z''], size: O(1) [0]

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