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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 423 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 99 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 264 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 63 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 360 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 139 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 320 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 82 ms]
    (48) CpxRNTS
    (49) FinalProof [FINISHED, 0 ms]
    (50) BOUNDS(1, n^1)
(51) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(52) TRS for Loop Detection
    (53) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (54) BEST
        (55) proven lower bound
            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
            (57) BOUNDS(n^1, INF)
        (58) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
mem(x, max(x)) -> not(null(x))

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

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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

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

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

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

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

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

(6)
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:

   f(x, nil) -> g(nil, x) [1]
   f(x, g(y, z)) -> g(f(x, y), z) [1]
   ++(x, nil) -> x [1]
   ++(x, g(y, z)) -> g(++(x, y), z) [1]
   null(nil) -> true [1]
   null(g(x, y)) -> false [1]
   mem(nil, y) -> false [1]
   mem(g(x, y), z) -> or(=(y, z), mem(x, z)) [1]
   max(g(g(nil, x), y)) -> max'(x, y) [1]
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x) [1]
   f(x, g(y, z)) -> g(f(x, y), z) [1]
   ++(x, nil) -> x [1]
   ++(x, g(y, z)) -> g(++(x, y), z) [1]
   null(nil) -> true [1]
   null(g(x, y)) -> false [1]
   mem(nil, y) -> false [1]
   mem(g(x, y), z) -> or(=(y, z), mem(x, z)) [1]
   max(g(g(nil, x), y)) -> max'(x, y) [1]
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u) [1]

The TRS has the following type information:
f :: max':u -> nil:g -> nil:g
nil :: nil:g
g :: nil:g -> max':u -> nil:g
++ :: nil:g -> nil:g -> nil:g
null :: nil:g -> true:false:or
true :: true:false:or
false :: true:false:or
mem :: nil:g -> a -> true:false:or
or :: = -> true:false:or -> true:false:or
= :: max':u -> a -> =
max :: nil:g -> max':u
max' :: max':u -> max':u -> max':u
u :: max':u

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

(9) 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:

   f_2
   ++_2
   null_1
   mem_2
   max_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(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:

   f(x, nil) -> g(nil, x) [1]
   f(x, g(y, z)) -> g(f(x, y), z) [1]
   ++(x, nil) -> x [1]
   ++(x, g(y, z)) -> g(++(x, y), z) [1]
   null(nil) -> true [1]
   null(g(x, y)) -> false [1]
   mem(nil, y) -> false [1]
   mem(g(x, y), z) -> or(=(y, z), mem(x, z)) [1]
   max(g(g(nil, x), y)) -> max'(x, y) [1]
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u) [1]

The TRS has the following type information:
f :: max':u -> nil:g -> nil:g
nil :: nil:g
g :: nil:g -> max':u -> nil:g
++ :: nil:g -> nil:g -> nil:g
null :: nil:g -> true:false:or
true :: true:false:or
false :: true:false:or
mem :: nil:g -> a -> true:false:or
or :: = -> true:false:or -> true:false:or
= :: max':u -> a -> =
max :: nil:g -> max':u
max' :: max':u -> max':u -> max':u
u :: max':u
const :: a
const1 :: =

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

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

(12)
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:

   f(x, nil) -> g(nil, x) [1]
   f(x, g(y, z)) -> g(f(x, y), z) [1]
   ++(x, nil) -> x [1]
   ++(x, g(y, z)) -> g(++(x, y), z) [1]
   null(nil) -> true [1]
   null(g(x, y)) -> false [1]
   mem(nil, y) -> false [1]
   mem(g(x, y), z) -> or(=(y, z), mem(x, z)) [1]
   max(g(g(nil, x), y)) -> max'(x, y) [1]
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u) [1]

The TRS has the following type information:
f :: max':u -> nil:g -> nil:g
nil :: nil:g
g :: nil:g -> max':u -> nil:g
++ :: nil:g -> nil:g -> nil:g
null :: nil:g -> true:false:or
true :: true:false:or
false :: true:false:or
mem :: nil:g -> a -> true:false:or
or :: = -> true:false:or -> true:false:or
= :: max':u -> a -> =
max :: nil:g -> max':u
max' :: max':u -> max':u -> max':u
u :: max':u
const :: a
const1 :: =

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

(13) 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
   true => 1
   false => 0
   u => 0
   const => 0
   const1 => 0

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

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

   ++(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(x, y) + z :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(x, y) + z :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + x :|: z'' = 0, z' = x, x >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   mem(z', z'') -{ 1 }-> 1 + (1 + y + z) + mem(x, z) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0


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

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

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

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0


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

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

   { mem }
   { max }
   { f }
   { null }
   { ++ }

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

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

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {mem}, {max}, {f}, {null}, {++}

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

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

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

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {mem}, {max}, {f}, {null}, {++}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {mem}, {max}, {f}, {null}, {++}
Previous analysis results are:
  mem: runtime: ?, size: O(n^2) [z' + z'*z'' + z'^2]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {max}, {f}, {null}, {++}
Previous analysis results are:
  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]

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

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

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

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {max}, {f}, {null}, {++}
Previous analysis results are:
  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}, {null}, {++}
Previous analysis results are:
  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']

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

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

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

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}, {null}, {++}
Previous analysis results are:
  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {null}, {++}
Previous analysis results are:
  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']

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

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {null}, {++}
Previous analysis results are:
  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {++}
Previous analysis results are:
  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]

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

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {++}
Previous analysis results are:
  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

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

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  ++: runtime: O(n^1) [1 + z''], size: O(n^1) [z' + z'']

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

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

(50)
BOUNDS(1, n^1)

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

(51) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(52)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

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

(53) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

++(x, g(y, z)) ->^+ g(++(x, y), z)

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [y / g(y, z)].

The result substitution is [ ].




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

(54)
Complex Obligation (BEST)

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

(55)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

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

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

(57)
BOUNDS(n^1, INF)

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

(58)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

S is empty.
Rewrite Strategy: FULL