/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^3), O(n^3))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3, n^3).

(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), 4 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 390 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 304 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 137 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 395 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) FinalProof [FINISHED, 0 ms]
    (36) BOUNDS(1, n^3)
(37) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(38) CpxTRS
    (39) SlicingProof [LOWER BOUND(ID), 0 ms]
    (40) CpxTRS
    (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) typed CpxTrs
    (43) OrderProof [LOWER BOUND(ID), 0 ms]
    (44) typed CpxTrs
    (45) RewriteLemmaProof [LOWER BOUND(ID), 308 ms]
    (46) BEST
        (47) proven lower bound
            (48) LowerBoundPropagationProof [FINISHED, 0 ms]
            (49) BOUNDS(n^1, INF)
        (50) typed CpxTrs
            (51) RewriteLemmaProof [LOWER BOUND(ID), 19 ms]
            (52) BEST
                (53) proven lower bound
                    (54) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (55) BOUNDS(n^2, INF)
                (56) typed CpxTrs
                    (57) RewriteLemmaProof [LOWER BOUND(ID), 4 ms]
                    (58) proven lower bound
                    (59) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (60) BOUNDS(n^3, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, 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: FULL
----------------------------------------

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (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: INNERMOST
----------------------------------------

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(4)
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
----------------------------------------

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

(6)
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
----------------------------------------

(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:

   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

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

(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, 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
----------------------------------------

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

(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:

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

(11) 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

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

(12)
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


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

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

(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


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

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

   { app }
   { reverse }
   { shuffle }

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

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

(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}

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

(19) 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'

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

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

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

(21) 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

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

(22)
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']

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

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

(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']

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

(25) 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

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

(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: {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]

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

(27) 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

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

(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) -{ 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]

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

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

(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]

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

(31) 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

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

(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: {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]

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

(33) 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

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

(34)
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]

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

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

(36)
BOUNDS(1, n^3)

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

(37) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(38)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


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: FULL
----------------------------------------

(39) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
add/0

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

(40)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

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

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

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

(42)
Obligation:
TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add

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

(43) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
app, reverse, shuffle

They will be analysed ascendingly in the following order:
app < reverse
reverse < shuffle

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

(44)
Obligation:
TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


The following defined symbols remain to be analysed:
app, reverse, shuffle

They will be analysed ascendingly in the following order:
app < reverse
reverse < shuffle

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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
app(gen_nil:add2_0(0), gen_nil:add2_0(b)) ->_R^Omega(1)
gen_nil:add2_0(b)

Induction Step:
app(gen_nil:add2_0(+(n4_0, 1)), gen_nil:add2_0(b)) ->_R^Omega(1)
add(app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b))) ->_IH
add(gen_nil:add2_0(+(b, c5_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(46)
Complex Obligation (BEST)

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

(47)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


The following defined symbols remain to be analysed:
app, reverse, shuffle

They will be analysed ascendingly in the following order:
app < reverse
reverse < shuffle

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

(48) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(49)
BOUNDS(n^1, INF)

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

(50)
Obligation:
TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


The following defined symbols remain to be analysed:
reverse, shuffle

They will be analysed ascendingly in the following order:
reverse < shuffle

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
reverse(gen_nil:add2_0(n439_0)) -> gen_nil:add2_0(n439_0), rt in Omega(1 + n439_0 + n439_0^2)

Induction Base:
reverse(gen_nil:add2_0(0)) ->_R^Omega(1)
nil

Induction Step:
reverse(gen_nil:add2_0(+(n439_0, 1))) ->_R^Omega(1)
app(reverse(gen_nil:add2_0(n439_0)), add(nil)) ->_IH
app(gen_nil:add2_0(c440_0), add(nil)) ->_L^Omega(1 + n439_0)
gen_nil:add2_0(+(n439_0, +(0, 1)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(52)
Complex Obligation (BEST)

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

(53)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


The following defined symbols remain to be analysed:
reverse, shuffle

They will be analysed ascendingly in the following order:
reverse < shuffle

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

(54) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(55)
BOUNDS(n^2, INF)

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

(56)
Obligation:
TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
reverse(gen_nil:add2_0(n439_0)) -> gen_nil:add2_0(n439_0), rt in Omega(1 + n439_0 + n439_0^2)


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


The following defined symbols remain to be analysed:
shuffle
----------------------------------------

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shuffle(gen_nil:add2_0(n623_0)) -> gen_nil:add2_0(n623_0), rt in Omega(1 + n623_0 + n623_0^2 + n623_0^3)

Induction Base:
shuffle(gen_nil:add2_0(0)) ->_R^Omega(1)
nil

Induction Step:
shuffle(gen_nil:add2_0(+(n623_0, 1))) ->_R^Omega(1)
add(shuffle(reverse(gen_nil:add2_0(n623_0)))) ->_L^Omega(1 + n623_0 + n623_0^2)
add(shuffle(gen_nil:add2_0(n623_0))) ->_IH
add(gen_nil:add2_0(c624_0))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(58)
Obligation:
Proved the lower bound n^3 for the following obligation:

TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
reverse(gen_nil:add2_0(n439_0)) -> gen_nil:add2_0(n439_0), rt in Omega(1 + n439_0 + n439_0^2)


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


The following defined symbols remain to be analysed:
shuffle
----------------------------------------

(59) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(60)
BOUNDS(n^3, INF)