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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(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), 4 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), 371 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 137 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 655 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 207 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 588 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^2)
(35) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CpxTRS
    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) typed CpxTrs
    (39) OrderProof [LOWER BOUND(ID), 0 ms]
    (40) typed CpxTrs
    (41) RewriteLemmaProof [LOWER BOUND(ID), 268 ms]
    (42) BEST
        (43) proven lower bound
            (44) LowerBoundPropagationProof [FINISHED, 0 ms]
            (45) BOUNDS(n^1, INF)
        (46) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
   select(Cons(x, xs)) -> selects(x, Nil, xs)
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
   selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil)
   select(Nil) -> Nil
   revapp(Nil, rest) -> rest

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

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, 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^2).


The TRS R consists of the following rules:

   selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs)) [1]
   select(Cons(x, xs)) -> selects(x, Nil, xs) [1]
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest)) [1]
   selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil) [1]
   select(Nil) -> Nil [1]
   revapp(Nil, rest) -> rest [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:

   selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs)) [1]
   select(Cons(x, xs)) -> selects(x, Nil, xs) [1]
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest)) [1]
   selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil) [1]
   select(Nil) -> Nil [1]
   revapp(Nil, rest) -> rest [1]

The TRS has the following type information:
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil

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:

   selects_3
   select_1
   revapp_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs)) [1]
   select(Cons(x, xs)) -> selects(x, Nil, xs) [1]
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest)) [1]
   selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil) [1]
   select(Nil) -> Nil [1]
   revapp(Nil, rest) -> rest [1]

The TRS has the following type information:
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil

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:

   selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs)) [1]
   select(Cons(x, xs)) -> selects(x, Nil, xs) [1]
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest)) [1]
   selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil) [1]
   select(Nil) -> Nil [1]
   revapp(Nil, rest) -> rest [1]

The TRS has the following type information:
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil

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

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

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

   revapp(z, z') -{ 1 }-> rest :|: z' = rest, rest >= 0, z = 0
   revapp(z, z') -{ 1 }-> revapp(xs, 1 + x + rest) :|: z = 1 + x + xs, xs >= 0, z' = rest, x >= 0, rest >= 0
   select(z) -{ 1 }-> selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + x + revapp(revprefix, 0)) + 0 :|: z'' = 0, z' = revprefix, x >= 0, revprefix >= 0, z = x
   selects(z, z', z'') -{ 1 }-> 1 + (1 + x' + revapp(revprefix, 1 + x + xs)) + selects(x, 1 + x' + revprefix, xs) :|: xs >= 0, z' = revprefix, x' >= 0, x >= 0, z'' = 1 + x + xs, revprefix >= 0, z = x'


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

(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:

   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   revapp(z, z') -{ 1 }-> revapp(xs, 1 + x + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   select(z) -{ 1 }-> selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 0)) + 0 :|: z'' = 0, z >= 0, z' >= 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 1 + x + xs)) + selects(x, 1 + z + z', xs) :|: xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0


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

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

   { revapp }
   { selects }
   { select }

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

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

   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   revapp(z, z') -{ 1 }-> revapp(xs, 1 + x + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   select(z) -{ 1 }-> selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 0)) + 0 :|: z'' = 0, z >= 0, z' >= 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 1 + x + xs)) + selects(x, 1 + z + z', xs) :|: xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0

Function symbols to be analyzed: {revapp}, {selects}, {select}

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

(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:

   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   revapp(z, z') -{ 1 }-> revapp(xs, 1 + x + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   select(z) -{ 1 }-> selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 0)) + 0 :|: z'' = 0, z >= 0, z' >= 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 1 + x + xs)) + selects(x, 1 + z + z', xs) :|: xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0

Function symbols to be analyzed: {revapp}, {selects}, {select}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: revapp
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:

   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   revapp(z, z') -{ 1 }-> revapp(xs, 1 + x + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   select(z) -{ 1 }-> selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 0)) + 0 :|: z'' = 0, z >= 0, z' >= 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 1 + x + xs)) + selects(x, 1 + z + z', xs) :|: xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0

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

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: revapp
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:

   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   revapp(z, z') -{ 1 }-> revapp(xs, 1 + x + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   select(z) -{ 1 }-> selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 0)) + 0 :|: z'' = 0, z >= 0, z' >= 0
   selects(z, z', z'') -{ 1 }-> 1 + (1 + z + revapp(z', 1 + x + xs)) + selects(x, 1 + z + z', xs) :|: xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0

Function symbols to be analyzed: {selects}, {select}
Previous analysis results are:
  revapp: 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:

   revapp(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   select(z) -{ 1 }-> selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 2 + z' }-> 1 + (1 + z + s) + selects(x, 1 + z + z', xs) :|: s >= 0, s <= z' + (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
   selects(z, z', z'') -{ 2 + z' }-> 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= z' + 0, z'' = 0, z >= 0, z' >= 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   revapp(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   select(z) -{ 1 }-> selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 2 + z' }-> 1 + (1 + z + s) + selects(x, 1 + z + z', xs) :|: s >= 0, s <= z' + (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
   selects(z, z', z'') -{ 2 + z' }-> 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= z' + 0, z'' = 0, z >= 0, z' >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   revapp(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   select(z) -{ 1 }-> selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 2 + z' }-> 1 + (1 + z + s) + selects(x, 1 + z + z', xs) :|: s >= 0, s <= z' + (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
   selects(z, z', z'') -{ 2 + z' }-> 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= z' + 0, z'' = 0, z >= 0, z' >= 0

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

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

(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:

   revapp(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   select(z) -{ 3 + x + x*xs + 2*xs + xs^2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 2 + 3 * xs + x * xs + 0 * xs + xs * xs, z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 5 + x + x*xs + 3*xs + xs*z + xs*z' + xs^2 + z + 2*z' }-> 1 + (1 + z + s) + s1 :|: s1 >= 0, s1 <= x + (1 + z + z') + 2 + 3 * xs + x * xs + (1 + z + z') * xs + xs * xs, s >= 0, s <= z' + (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
   selects(z, z', z'') -{ 2 + z' }-> 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= z' + 0, z'' = 0, z >= 0, z' >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: select
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   revapp(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   select(z) -{ 3 + x + x*xs + 2*xs + xs^2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 2 + 3 * xs + x * xs + 0 * xs + xs * xs, z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 5 + x + x*xs + 3*xs + xs*z + xs*z' + xs^2 + z + 2*z' }-> 1 + (1 + z + s) + s1 :|: s1 >= 0, s1 <= x + (1 + z + z') + 2 + 3 * xs + x * xs + (1 + z + z') * xs + xs * xs, s >= 0, s <= z' + (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
   selects(z, z', z'') -{ 2 + z' }-> 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= z' + 0, z'' = 0, z >= 0, z' >= 0

Function symbols to be analyzed: {select}
Previous analysis results are:
  revapp: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  selects: runtime: O(n^2) [2 + z + z*z'' + z' + z'*z'' + 2*z'' + z''^2], size: O(n^2) [2 + z + z*z'' + z' + z'*z'' + 3*z'' + z''^2]
  select: runtime: ?, size: INF

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   revapp(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   revapp(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   select(z) -{ 3 + x + x*xs + 2*xs + xs^2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 2 + 3 * xs + x * xs + 0 * xs + xs * xs, z = 1 + x + xs, xs >= 0, x >= 0
   select(z) -{ 1 }-> 0 :|: z = 0
   selects(z, z', z'') -{ 5 + x + x*xs + 3*xs + xs*z + xs*z' + xs^2 + z + 2*z' }-> 1 + (1 + z + s) + s1 :|: s1 >= 0, s1 <= x + (1 + z + z') + 2 + 3 * xs + x * xs + (1 + z + z') * xs + xs * xs, s >= 0, s <= z' + (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
   selects(z, z', z'') -{ 2 + z' }-> 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= z' + 0, z'' = 0, z >= 0, z' >= 0

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

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

(33) FinalProof (FINISHED)
Computed overall runtime complexity
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(34)
BOUNDS(1, n^2)

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(35) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(36)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
   select(Cons(x, xs)) -> selects(x, Nil, xs)
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
   selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil)
   select(Nil) -> Nil
   revapp(Nil, rest) -> rest

S is empty.
Rewrite Strategy: INNERMOST
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(37) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(38)
Obligation:
Innermost TRS:
Rules:
selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) -> selects(x, Nil, xs)
revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) -> Nil
revapp(Nil, rest) -> rest

Types:
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil

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(39) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
selects, revapp

They will be analysed ascendingly in the following order:
revapp < selects

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(40)
Obligation:
Innermost TRS:
Rules:
selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) -> selects(x, Nil, xs)
revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) -> Nil
revapp(Nil, rest) -> rest

Types:
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
revapp, selects

They will be analysed ascendingly in the following order:
revapp < selects

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(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
revapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) -> gen_Cons:Nil2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
revapp(gen_Cons:Nil2_0(0), gen_Cons:Nil2_0(b)) ->_R^Omega(1)
gen_Cons:Nil2_0(b)

Induction Step:
revapp(gen_Cons:Nil2_0(+(n4_0, 1)), gen_Cons:Nil2_0(b)) ->_R^Omega(1)
revapp(gen_Cons:Nil2_0(n4_0), Cons(Nil, gen_Cons:Nil2_0(b))) ->_IH
gen_Cons:Nil2_0(+(+(b, 1), c5_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(42)
Complex Obligation (BEST)

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(43)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) -> selects(x, Nil, xs)
revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) -> Nil
revapp(Nil, rest) -> rest

Types:
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
revapp, selects

They will be analysed ascendingly in the following order:
revapp < selects

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(44) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(45)
BOUNDS(n^1, INF)

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(46)
Obligation:
Innermost TRS:
Rules:
selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) -> selects(x, Nil, xs)
revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) -> Nil
revapp(Nil, rest) -> rest

Types:
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Lemmas:
revapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) -> gen_Cons:Nil2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
selects