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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 157 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
(5) CpxWeightedTrsRenamingProof [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), 0 ms]
(18) CpxRNTS
(19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 353 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), 149 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 994 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 480 ms]
(36) CpxRNTS
(37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
(42) CpxRNTS
(43) FinalProof [FINISHED, 0 ms]
(44) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](<(x', x), Cons(x', Cons(x, xs)))
   ordered(Cons(x, Nil)) -> True
   ordered(Nil) -> True
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> ordered(xs)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   ordered[Ite](True, Cons(x, xs)) -> ordered(xs)
   ordered[Ite](False, xs) -> False

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](<(x', x), Cons(x', Cons(x, xs)))
   ordered(Cons(x, Nil)) -> True
   ordered(Nil) -> True
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> ordered(xs)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   ordered[Ite](True, Cons(x, xs)) -> ordered(xs)
   ordered[Ite](False, xs) -> False

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^1).


The TRS R consists of the following rules:

   ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](<(x', x), Cons(x', Cons(x, xs))) [1]
   ordered(Cons(x, Nil)) -> True [1]
   ordered(Nil) -> True [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> ordered(xs) [1]
   <(S(x), S(y)) -> <(x, y) [0]
   <(0, S(y)) -> True [0]
   <(x, 0) -> False [0]
   ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0]
   ordered[Ite](False, xs) -> False [0]

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

(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

< => lt

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

(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:

   ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](lt(x', x), Cons(x', Cons(x, xs))) [1]
   ordered(Cons(x, Nil)) -> True [1]
   ordered(Nil) -> True [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> ordered(xs) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0]
   ordered[Ite](False, xs) -> False [0]

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:

   ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](lt(x', x), Cons(x', Cons(x, xs))) [1]
   ordered(Cons(x, Nil)) -> True [1]
   ordered(Nil) -> True [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> ordered(xs) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0]
   ordered[Ite](False, xs) -> False [0]

The TRS has the following type information:
ordered :: Cons:Nil -> True:False
Cons :: S:0 -> Cons:Nil -> Cons:Nil
ordered[Ite] :: True:False -> Cons:Nil -> True:False
lt :: S:0 -> S:0 -> True:False
Nil :: Cons:Nil
True :: True:False
notEmpty :: Cons:Nil -> True:False
False :: True:False
goal :: Cons:Nil -> True:False
S :: S:0 -> S:0
0 :: S:0

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:

   ordered_1
   notEmpty_1
   goal_1

(c) The following functions are completely defined:

   lt_2
   ordered[Ite]_2

Due to the following rules being added:

   lt(v0, v1) -> null_lt [0]
   ordered[Ite](v0, v1) -> null_ordered[Ite] [0]

And the following fresh constants:    null_lt, null_ordered[Ite]

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

(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:

   ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](lt(x', x), Cons(x', Cons(x, xs))) [1]
   ordered(Cons(x, Nil)) -> True [1]
   ordered(Nil) -> True [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> ordered(xs) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0]
   ordered[Ite](False, xs) -> False [0]
   lt(v0, v1) -> null_lt [0]
   ordered[Ite](v0, v1) -> null_ordered[Ite] [0]

The TRS has the following type information:
ordered :: Cons:Nil -> True:False:null_lt:null_ordered[Ite]
Cons :: S:0 -> Cons:Nil -> Cons:Nil
ordered[Ite] :: True:False:null_lt:null_ordered[Ite] -> Cons:Nil -> True:False:null_lt:null_ordered[Ite]
lt :: S:0 -> S:0 -> True:False:null_lt:null_ordered[Ite]
Nil :: Cons:Nil
True :: True:False:null_lt:null_ordered[Ite]
notEmpty :: Cons:Nil -> True:False:null_lt:null_ordered[Ite]
False :: True:False:null_lt:null_ordered[Ite]
goal :: Cons:Nil -> True:False:null_lt:null_ordered[Ite]
S :: S:0 -> S:0
0 :: S:0
null_lt :: True:False:null_lt:null_ordered[Ite]
null_ordered[Ite] :: True:False:null_lt:null_ordered[Ite]

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:

   ordered(Cons(S(x''), Cons(S(y'), xs))) -> ordered[Ite](lt(x'', y'), Cons(S(x''), Cons(S(y'), xs))) [1]
   ordered(Cons(0, Cons(S(y''), xs))) -> ordered[Ite](True, Cons(0, Cons(S(y''), xs))) [1]
   ordered(Cons(x', Cons(0, xs))) -> ordered[Ite](False, Cons(x', Cons(0, xs))) [1]
   ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](null_lt, Cons(x', Cons(x, xs))) [1]
   ordered(Cons(x, Nil)) -> True [1]
   ordered(Nil) -> True [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> ordered(xs) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0]
   ordered[Ite](False, xs) -> False [0]
   lt(v0, v1) -> null_lt [0]
   ordered[Ite](v0, v1) -> null_ordered[Ite] [0]

The TRS has the following type information:
ordered :: Cons:Nil -> True:False:null_lt:null_ordered[Ite]
Cons :: S:0 -> Cons:Nil -> Cons:Nil
ordered[Ite] :: True:False:null_lt:null_ordered[Ite] -> Cons:Nil -> True:False:null_lt:null_ordered[Ite]
lt :: S:0 -> S:0 -> True:False:null_lt:null_ordered[Ite]
Nil :: Cons:Nil
True :: True:False:null_lt:null_ordered[Ite]
notEmpty :: Cons:Nil -> True:False:null_lt:null_ordered[Ite]
False :: True:False:null_lt:null_ordered[Ite]
goal :: Cons:Nil -> True:False:null_lt:null_ordered[Ite]
S :: S:0 -> S:0
0 :: S:0
null_lt :: True:False:null_lt:null_ordered[Ite]
null_ordered[Ite] :: True:False:null_lt:null_ordered[Ite]

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 => 2
   False => 1
   0 => 0
   null_lt => 0
   null_ordered[Ite] => 0

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

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

   goal(z) -{ 1 }-> ordered(xs) :|: xs >= 0, z = xs
   lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: x >= 0, z = 1 + x + 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: xs >= 0, z = 1, z' = xs
   ordered[Ite](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

(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:

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { lt }
   { notEmpty }
   { ordered[Ite], ordered }
   { goal }

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

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

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {notEmpty}, {ordered[Ite],ordered}, {goal}

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

(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:

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {notEmpty}, {ordered[Ite],ordered}, {goal}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {notEmpty}, {ordered[Ite],ordered}, {goal}
Previous analysis results are:
  lt: runtime: ?, size: O(1) [2]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: lt
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {notEmpty}, {ordered[Ite],ordered}, {goal}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [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:

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {notEmpty}, {ordered[Ite],ordered}, {goal}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {notEmpty}, {ordered[Ite],ordered}, {goal}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: ?, size: O(1) [2]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {ordered[Ite],ordered}, {goal}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]

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

(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:

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {ordered[Ite],ordered}, {goal}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ordered[Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

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

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {ordered[Ite],ordered}, {goal}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  ordered[Ite]: runtime: ?, size: O(1) [2]
  ordered: runtime: ?, size: O(1) [2]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: ordered[Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + z'

Computed RUNTIME bound using CoFloCo for: ordered
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + z

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

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

   goal(z) -{ 1 }-> ordered(z) :|: z >= 0
   lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  ordered[Ite]: runtime: O(n^1) [4 + z'], size: O(1) [2]
  ordered: runtime: O(n^1) [5 + z], size: O(1) [2]

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

(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:

   goal(z) -{ 6 + z }-> s4 :|: s4 >= 0, s4 <= 2, z >= 0
   lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 9 + x'' + xs + y' }-> s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 8 + xs + y'' }-> s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 7 + x' + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 7 + x + x' + xs }-> s3 :|: s3 >= 0, s3 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 5 + xs }-> s5 :|: s5 >= 0, s5 <= 2, z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  ordered[Ite]: runtime: O(n^1) [4 + z'], size: O(1) [2]
  ordered: runtime: O(n^1) [5 + z], size: O(1) [2]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z) -{ 6 + z }-> s4 :|: s4 >= 0, s4 <= 2, z >= 0
   lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 9 + x'' + xs + y' }-> s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 8 + xs + y'' }-> s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 7 + x' + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 7 + x + x' + xs }-> s3 :|: s3 >= 0, s3 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 5 + xs }-> s5 :|: s5 >= 0, s5 <= 2, z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  ordered[Ite]: runtime: O(n^1) [4 + z'], size: O(1) [2]
  ordered: runtime: O(n^1) [5 + z], size: O(1) [2]
  goal: runtime: ?, size: O(1) [2]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 6 + z

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

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

   goal(z) -{ 6 + z }-> s4 :|: s4 >= 0, s4 <= 2, z >= 0
   lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   ordered(z) -{ 9 + x'' + xs + y' }-> s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0
   ordered(z) -{ 8 + xs + y'' }-> s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0
   ordered(z) -{ 7 + x' + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs)
   ordered(z) -{ 7 + x + x' + xs }-> s3 :|: s3 >= 0, s3 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0
   ordered(z) -{ 1 }-> 2 :|: z = 0
   ordered[Ite](z, z') -{ 5 + xs }-> s5 :|: s5 >= 0, s5 <= 2, z = 2, xs >= 0, z' = 1 + x + xs, x >= 0
   ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1
   ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  ordered[Ite]: runtime: O(n^1) [4 + z'], size: O(1) [2]
  ordered: runtime: O(n^1) [5 + z], size: O(1) [2]
  goal: runtime: O(n^1) [6 + z], size: O(1) [2]

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

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

(44)
BOUNDS(1, n^1)