WORST_CASE(?,O(n^1))
proof of input_B5sEyI8DQe.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) 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), 0 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), 232 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 90 ms]
(20) CpxRNTS
(21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 122 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
(26) CpxRNTS
(27) FinalProof [FINISHED, 0 ms]
(28) 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:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

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:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1]
   duplicate(Nil) -> Nil [1]
   goal(x) -> duplicate(x) [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:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1]
   duplicate(Nil) -> Nil [1]
   goal(x) -> duplicate(x) [1]

The TRS has the following type information:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil

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:

   duplicate_1
   goal_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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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:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1]
   duplicate(Nil) -> Nil [1]
   goal(x) -> duplicate(x) [1]

The TRS has the following type information:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
const :: a

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:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1]
   duplicate(Nil) -> Nil [1]
   goal(x) -> duplicate(x) [1]

The TRS has the following type information:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
const :: a

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:

   Nil => 0
   const => 0

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

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 1 }-> duplicate(x) :|: x >= 0, z = x


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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:

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 1 }-> duplicate(z) :|: z >= 0


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

   { duplicate }
   { goal }

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

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 1 }-> duplicate(z) :|: z >= 0

Function symbols to be analyzed: {duplicate}, {goal}

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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:

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 1 }-> duplicate(z) :|: z >= 0

Function symbols to be analyzed: {duplicate}, {goal}

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(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 1 }-> duplicate(z) :|: z >= 0

Function symbols to be analyzed: {duplicate}, {goal}
Previous analysis results are:
  duplicate: runtime: ?, size: O(n^1) [2*z]

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(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: duplicate
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:

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 1 }-> duplicate(z) :|: z >= 0

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

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(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(22)
Obligation:
Complexity RNTS consisting of the following rules:

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 2 + xs }-> 1 + x + (1 + x + s) :|: s >= 0, s <= 2 * xs, z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * z, z >= 0

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

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(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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(24)
Obligation:
Complexity RNTS consisting of the following rules:

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 2 + xs }-> 1 + x + (1 + x + s) :|: s >= 0, s <= 2 * xs, z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * z, z >= 0

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

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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(26)
Obligation:
Complexity RNTS consisting of the following rules:

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 2 + xs }-> 1 + x + (1 + x + s) :|: s >= 0, s <= 2 * xs, z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * z, z >= 0

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

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(27) FinalProof (FINISHED)
Computed overall runtime complexity
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(28)
BOUNDS(1, n^1)