The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be 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:

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: INNERMOST

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed relative TRS to weighted TRS

(2) 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

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) 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

(5) CompletionProof (UPPER BOUND(ID) transformation)

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
goal

(c) The following functions are completely defined:

lte
even
and

Due to the following rules being added:

and(v0, v1) → null_and [0]

And the following fresh constants:

null_and, const

(6) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

(7) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(8) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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

(10) 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

(11) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

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

(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(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

(13) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(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 :|: 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

(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

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

(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

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

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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}
Previous analysis results are:

and: runtime: ?, size: O(1) [2]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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: {notEmpty}, {lte}, {even}, {goal}
Previous analysis results are:

and: runtime: O(1) [0], size: O(1) [2]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(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: {notEmpty}, {lte}, {even}, {goal}
Previous analysis results are:

and: runtime: O(1) [0], size: O(1) [2]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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] notEmpty: runtime: ?, size: O(1) [2]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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: {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]

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(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: {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]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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] lte: runtime: ?, size: O(1) [2]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: lte
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'

(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: {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(n1) [1 + z'], size: O(1) [2]

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(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') -{ 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(n1) [1 + z'], size: O(1) [2]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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') -{ 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(n1) [1 + z'], size: O(1) [2] even: runtime: ?, size: O(1) [2]

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: even
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z

(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: {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(n1) [1 + z'], size: O(1) [2] even: runtime: O(n1) [1 + z], size: O(1) [2]

(39) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(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) -{ 2 + 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') -{ 5 + 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') -{ 4 + 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(n1) [1 + z'], size: O(1) [2] even: runtime: O(n1) [1 + z], size: O(1) [2]

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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) -{ 2 + 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') -{ 5 + 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') -{ 4 + 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(n1) [1 + z'], size: O(1) [2] even: runtime: O(n1) [1 + z], size: O(1) [2] goal: runtime: ?, size: O(1) [2]

(43) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + z + z'

(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) -{ 2 + 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') -{ 5 + 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') -{ 4 + 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(n1) [1 + z'], size: O(1) [2] even: runtime: O(n1) [1 + z], size: O(1) [2] goal: runtime: O(n1) [3 + z + z'], size: O(1) [2]

(45) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(46) BOUNDS(1, n^1)