WORST_CASE(?,O(n^3))
proof of input_vST2183VuA.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^3).

(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), 11 ms]
(10) CpxRNTS
(11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxRNTS
(13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms]
(14) CpxRNTS
(15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) IntTrsBoundProof [UPPER BOUND(ID), 387 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 111 ms]
(20) CpxRNTS
(21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 413 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
(26) CpxRNTS
(27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 385 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 16 ms]
(32) CpxRNTS
(33) FinalProof [FINISHED, 0 ms]
(34) BOUNDS(1, n^3)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(nil) -> nil
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(nil) -> nil [1]
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(nil) -> nil [1]
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1]

The TRS has the following type information:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: a -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add

Rewrite Strategy: INNERMOST
----------------------------------------

(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:

   shuffle_1

(c) The following functions are completely defined:

   reverse_1
   app_2

Due to the following rules being added:
none

And the following fresh constants:    const

----------------------------------------

(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:

   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(nil) -> nil [1]
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1]

The TRS has the following type information:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: a -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
const :: a

Rewrite Strategy: INNERMOST
----------------------------------------

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

(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:

   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, nil)) -> app(nil, add(n, nil)) [2]
   reverse(add(n, add(n', x'))) -> app(app(reverse(x'), add(n', nil)), add(n, nil)) [2]
   shuffle(nil) -> nil [1]
   shuffle(add(n, nil)) -> add(n, shuffle(nil)) [2]
   shuffle(add(n, add(n'', x''))) -> add(n, shuffle(app(reverse(x''), add(n'', nil)))) [2]

The TRS has the following type information:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: a -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
const :: a

Rewrite Strategy: INNERMOST
----------------------------------------

(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

----------------------------------------

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

   app(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y
   reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 2 }-> app(0, 1 + n + 0) :|: z = 1 + n + 0, n >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + n + shuffle(0) :|: z = 1 + n + 0, n >= 0


----------------------------------------

(11) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0


----------------------------------------

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

   { app }
   { reverse }
   { shuffle }

----------------------------------------

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {reverse}, {shuffle}

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(15) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {reverse}, {shuffle}

----------------------------------------

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

----------------------------------------

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

----------------------------------------

(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

----------------------------------------

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

----------------------------------------

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: reverse
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 3*z + 2*z^2

----------------------------------------

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

----------------------------------------

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 8 + s'' + s1 + 3*x' + 2*x'^2 }-> s2 :|: s'' >= 0, s'' <= x', s1 >= 0, s1 <= s'' + (1 + n' + 0), s2 >= 0, s2 <= s1 + (1 + n + 0), n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 7 + s3 + 3*x'' + 2*x''^2 }-> 1 + n + shuffle(s4) :|: s3 >= 0, s3 <= x'', s4 >= 0, s4 <= s3 + (1 + n'' + 0), n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 8 + s'' + s1 + 3*x' + 2*x'^2 }-> s2 :|: s'' >= 0, s'' <= x', s1 >= 0, s1 <= s'' + (1 + n' + 0), s2 >= 0, s2 <= s1 + (1 + n + 0), n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 7 + s3 + 3*x'' + 2*x''^2 }-> 1 + n + shuffle(s4) :|: s3 >= 0, s3 <= x'', s4 >= 0, s4 <= s3 + (1 + n'' + 0), n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

Function symbols to be analyzed: {shuffle}
Previous analysis results are:
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  reverse: runtime: O(n^2) [4 + 3*z + 2*z^2], size: O(n^1) [z]
  shuffle: runtime: ?, size: O(n^1) [z]

----------------------------------------

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: shuffle
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 9*z + 4*z^2 + 2*z^3

----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 8 + s'' + s1 + 3*x' + 2*x'^2 }-> s2 :|: s'' >= 0, s'' <= x', s1 >= 0, s1 <= s'' + (1 + n' + 0), s2 >= 0, s2 <= s1 + (1 + n + 0), n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 7 + s3 + 3*x'' + 2*x''^2 }-> 1 + n + shuffle(s4) :|: s3 >= 0, s3 <= x'', s4 >= 0, s4 <= s3 + (1 + n'' + 0), n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
   shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  reverse: runtime: O(n^2) [4 + 3*z + 2*z^2], size: O(n^1) [z]
  shuffle: runtime: O(n^3) [1 + 9*z + 4*z^2 + 2*z^3], size: O(n^1) [z]

----------------------------------------

(33) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(34)
BOUNDS(1, n^3)