WORST_CASE(?,O(n^1))
proof of input_gmZ9TGmqqB.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 136 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxWeightedTrs
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTypedWeightedTrs
(7) CompletionProof [UPPER BOUND(ID), 0 ms]
(8) CpxTypedWeightedCompleteTrs
(9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedCompleteTrs
(11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
(12) CpxRNTS
(13) InliningProof [UPPER BOUND(ID), 276 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), 264 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 56 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 151 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 396 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 119 ms]
(36) CpxRNTS
(37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 251 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
(42) CpxRNTS
(43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 447 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 164 ms]
(48) CpxRNTS
(49) FinalProof [FINISHED, 0 ms]
(50) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs)
   lte(Cons(x, xs), Nil) -> False
   even(Cons(x, Nil)) -> False
   even(Cons(x', Cons(x, xs))) -> even(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   lte(Nil, y) -> True
   even(Nil) -> True
   goal(x, y) -> and(lte(x, y), even(x))

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs)
   lte(Cons(x, xs), Nil) -> False
   even(Cons(x, Nil)) -> False
   even(Cons(x', Cons(x, xs))) -> even(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   lte(Nil, y) -> True
   even(Nil) -> True
   goal(x, y) -> and(lte(x, y), even(x))

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToWeightedTrsProof (UPPER BOUND(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:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1]
   lte(Cons(x, xs), Nil) -> False [1]
   even(Cons(x, Nil)) -> False [1]
   even(Cons(x', Cons(x, xs))) -> even(xs) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   lte(Nil, y) -> True [1]
   even(Nil) -> True [1]
   goal(x, y) -> and(lte(x, y), even(x)) [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1]
   lte(Cons(x, xs), Nil) -> False [1]
   even(Cons(x, Nil)) -> False [1]
   even(Cons(x', Cons(x, xs))) -> even(xs) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   lte(Nil, y) -> True [1]
   even(Nil) -> True [1]
   goal(x, y) -> and(lte(x, y), even(x)) [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]

The TRS has the following type information:
lte :: Cons:Nil -> Cons:Nil -> False:True
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
and :: False:True -> False:True -> False:True

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

(7) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   notEmpty_1
   goal_2

(c) The following functions are completely defined:

   lte_2
   even_1
   and_2

Due to the following rules being added:

   and(v0, v1) -> null_and [0]

And the following fresh constants:    null_and, const

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

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1]
   lte(Cons(x, xs), Nil) -> False [1]
   even(Cons(x, Nil)) -> False [1]
   even(Cons(x', Cons(x, xs))) -> even(xs) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   lte(Nil, y) -> True [1]
   even(Nil) -> True [1]
   goal(x, y) -> and(lte(x, y), even(x)) [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
lte :: Cons:Nil -> Cons:Nil -> False:True:null_and
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True:null_and
even :: Cons:Nil -> False:True:null_and
notEmpty :: Cons:Nil -> False:True:null_and
True :: False:True:null_and
goal :: Cons:Nil -> Cons:Nil -> False:True:null_and
and :: False:True:null_and -> False:True:null_and -> False:True:null_and
null_and :: False:True:null_and
const :: a

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1]
   lte(Cons(x, xs), Nil) -> False [1]
   even(Cons(x, Nil)) -> False [1]
   even(Cons(x', Cons(x, xs))) -> even(xs) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   lte(Nil, y) -> True [1]
   even(Nil) -> True [1]
   goal(Cons(x'', Nil), Cons(x1, xs1)) -> and(lte(Nil, xs1), False) [3]
   goal(Cons(x'', Cons(x3, xs3)), Cons(x1, xs1)) -> and(lte(Cons(x3, xs3), xs1), even(xs3)) [3]
   goal(Cons(x2, Nil), Nil) -> and(False, False) [3]
   goal(Cons(x2, Cons(x4, xs4)), Nil) -> and(False, even(xs4)) [3]
   goal(Nil, y) -> and(True, True) [3]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
lte :: Cons:Nil -> Cons:Nil -> False:True:null_and
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True:null_and
even :: Cons:Nil -> False:True:null_and
notEmpty :: Cons:Nil -> False:True:null_and
True :: False:True:null_and
goal :: Cons:Nil -> Cons:Nil -> False:True:null_and
and :: False:True:null_and -> False:True:null_and -> False:True:null_and
null_and :: False:True:null_and
const :: a

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

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   False => 1
   True => 2
   null_and => 0
   const => 0

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

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

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x + 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, x'' >= 0, z = 1 + x'' + 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(2, 2) :|: y >= 0, z = 0, z' = y
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> and(1, 1) :|: z = 1 + x2 + 0, x2 >= 0, z' = 0
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0


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

(13) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1

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

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

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x + 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, x'' >= 0, z = 1 + x'' + 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: y >= 0, z = 0, z' = y, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z = 1 + x2 + 0, x2 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z = 1 + x2 + 0, x2 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: y >= 0, z = 0, z' = y, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0


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

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

   { and }
   { notEmpty }
   { lte }
   { even }
   { goal }

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

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

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {and}, {notEmpty}, {lte}, {even}, {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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {and}, {notEmpty}, {lte}, {even}, {goal}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: and
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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: and
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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {lte}, {even}, {goal}
Previous analysis results are:
  and: 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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {lte}, {even}, {goal}
Previous analysis results are:
  and: 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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {lte}, {even}, {goal}
Previous analysis results are:
  and: 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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {lte}, {even}, {goal}
Previous analysis results are:
  and: 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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {lte}, {even}, {goal}
Previous analysis results are:
  and: 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: lte
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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: lte
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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {even}, {goal}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  lte: runtime: O(n^1) [1 + 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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 4 + xs1 }-> and(s1, even(xs3)) :|: s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {even}, {goal}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  lte: runtime: O(n^1) [1 + z'], size: O(1) [2]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: even
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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 4 + xs1 }-> and(s1, even(xs3)) :|: s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {even}, {goal}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  lte: runtime: O(n^1) [1 + z'], size: O(1) [2]
  even: runtime: ?, size: O(1) [2]

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(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: even
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z

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

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

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 4 + xs1 }-> and(s1, even(xs3)) :|: s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  lte: runtime: O(n^1) [1 + z'], size: O(1) [2]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]

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(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:

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 3 + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 6 + xs1 + xs3 }-> s4 :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 5 + xs4 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  lte: runtime: O(n^1) [1 + z'], size: O(1) [2]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]

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

(45) 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

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

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

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 3 + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 6 + xs1 + xs3 }-> s4 :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 5 + xs4 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  lte: runtime: O(n^1) [1 + z'], size: O(1) [2]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]
  goal: runtime: ?, size: O(1) [2]

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

(47) 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: 3 + z + z'

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

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

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   even(z) -{ 3 + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: z - 1 >= 0
   goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1
   goal(z, z') -{ 6 + xs1 + xs3 }-> s4 :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0
   goal(z, z') -{ 5 + xs4 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0
   goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2
   goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1
   goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0

Function symbols to be analyzed: 
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  lte: runtime: O(n^1) [1 + z'], size: O(1) [2]
  even: runtime: O(n^1) [2 + z], size: O(1) [2]
  goal: runtime: O(n^1) [3 + z + z'], size: O(1) [2]

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

(50)
BOUNDS(1, n^1)