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

(0) Obligation:

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


The TRS R consists of the following rules:

naiverev(Cons(x, xs)) → app(naiverev(xs), Cons(x, Nil))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(2) Obligation:

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


The TRS R consists of the following rules:

naiverev(Cons(x, xs)) → app(naiverev(xs), Cons(x, Nil)) [1]
app(Cons(x, xs), ys) → Cons(x, app(xs, ys)) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
naiverev(Nil) → Nil [1]
app(Nil, ys) → ys [1]
goal(xs) → naiverev(xs) [1]

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:

naiverev(Cons(x, xs)) → app(naiverev(xs), Cons(x, Nil)) [1]
app(Cons(x, xs), ys) → Cons(x, app(xs, ys)) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
naiverev(Nil) → Nil [1]
app(Nil, ys) → ys [1]
goal(xs) → naiverev(xs) [1]

The TRS has the following type information:

naiverev :: Cons:Nil → Cons:Nil Cons :: a → Cons:Nil → Cons:Nil app :: Cons:Nil → Cons:Nil → Cons:Nil Nil :: Cons:Nil notEmpty :: Cons:Nil → True:False True :: True:False False :: True:False goal :: Cons:Nil → Cons:Nil

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:

naiverev
app

Due to the following rules being added:
none

And the following fresh constants:

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: naiverev(Cons(x, xs)) → app(naiverev(xs), Cons(x, Nil)) [1] app(Cons(x, xs), ys) → Cons(x, app(xs, ys)) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] naiverev(Nil) → Nil [1] app(Nil, ys) → ys [1] goal(xs) → naiverev(xs) [1] The TRS has the following type information: naiverev :: Cons:Nil → Cons:Nil Cons :: a → Cons:Nil → Cons:Nil app :: Cons:Nil → Cons:Nil → Cons:Nil Nil :: Cons:Nil notEmpty :: Cons:Nil → True:False True :: True:False False :: True:False goal :: Cons:Nil → Cons:Nil 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: naiverev(Cons(x, Cons(x', xs'))) → app(app(naiverev(xs'), Cons(x', Nil)), Cons(x, Nil)) [2] naiverev(Cons(x, Nil)) → app(Nil, Cons(x, Nil)) [2] app(Cons(x, xs), ys) → Cons(x, app(xs, ys)) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] naiverev(Nil) → Nil [1] app(Nil, ys) → ys [1] goal(xs) → naiverev(xs) [1] The TRS has the following type information: naiverev :: Cons:Nil → Cons:Nil Cons :: a → Cons:Nil → Cons:Nil app :: Cons:Nil → Cons:Nil → Cons:Nil Nil :: Cons:Nil notEmpty :: Cons:Nil → True:False True :: True:False False :: True:False goal :: Cons:Nil → Cons:Nil 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
True => 1
False => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ ys :|: z' = ys, ys >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(xs) :|: xs >= 0, z = xs
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 2 }→ app(0, 1 + x + 0) :|: x >= 0, z = 1 + x + 0
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

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

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 2 }→ app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

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

Found the following analysis order by SCC decomposition:

{ notEmpty } { app } { naiverev } { goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 2 }→ app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {app}, {naiverev}, {goal}

(15) 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: 1

(16) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 2 }→ app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {app}, {naiverev}, {goal}
Previous analysis results are:

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

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 2 }→ app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {app}, {naiverev}, {goal}
Previous analysis results are:

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

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 2 }→ app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {app}, {naiverev}, {goal}
Previous analysis results are:

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

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 2 }→ app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {app}, {naiverev}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] app: runtime: ?, size: O(n1) [z + z']

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 2 }→ app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {naiverev}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n1) [1 + z], size: O(n1) [z + z']

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + xs }→ 1 + x + s' :|: s' >= 0, s' <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {naiverev}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n1) [1 + z], size: O(n1) [z + z']

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: naiverev
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(28) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + xs }→ 1 + x + s' :|: s' >= 0, s' <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {naiverev}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n1) [1 + z], size: O(n1) [z + z'] naiverev: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using CoFloCo for: naiverev
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 4 + 3·z + 2·z²

(30) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + xs }→ 1 + x + s' :|: s' >= 0, s' <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ naiverev(z) :|: z >= 0
naiverev(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
naiverev(z) -{ 2 }→ app(app(naiverev(xs'), 1 + x' + 0), 1 + x + 0) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n1) [1 + z], size: O(n1) [z + z'] naiverev: runtime: O(n2) [4 + 3·z + 2·z2], size: O(n1) [z]

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + xs }→ 1 + x + s' :|: s' >= 0, s' <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 5 + 3·z + 2·z² }→ s3 :|: s3 >= 0, s3 <= 1 * z, z >= 0
naiverev(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
naiverev(z) -{ 8 + s'' + s1 + 3·xs' + 2·xs'² }→ s2 :|: s'' >= 0, s'' <= 1 * xs', s1 >= 0, s1 <= 1 * s'' + 1 * (1 + x' + 0), s2 >= 0, s2 <= 1 * s1 + 1 * (1 + x + 0), x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n1) [1 + z], size: O(n1) [z + z'] naiverev: runtime: O(n2) [4 + 3·z + 2·z2], size: O(n1) [z]

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + xs }→ 1 + x + s' :|: s' >= 0, s' <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 5 + 3·z + 2·z² }→ s3 :|: s3 >= 0, s3 <= 1 * z, z >= 0
naiverev(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
naiverev(z) -{ 8 + s'' + s1 + 3·xs' + 2·xs'² }→ s2 :|: s'' >= 0, s'' <= 1 * xs', s1 >= 0, s1 <= 1 * s'' + 1 * (1 + x' + 0), s2 >= 0, s2 <= 1 * s1 + 1 * (1 + x + 0), x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n1) [1 + z], size: O(n1) [z + z'] naiverev: runtime: O(n2) [4 + 3·z + 2·z2], size: O(n1) [z] goal: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using KoAT for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 5 + 3·z + 2·z²

(36) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + xs }→ 1 + x + s' :|: s' >= 0, s' <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 5 + 3·z + 2·z² }→ s3 :|: s3 >= 0, s3 <= 1 * z, z >= 0
naiverev(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
naiverev(z) -{ 8 + s'' + s1 + 3·xs' + 2·xs'² }→ s2 :|: s'' >= 0, s'' <= 1 * xs', s1 >= 0, s1 <= 1 * s'' + 1 * (1 + x' + 0), s2 >= 0, s2 <= 1 * s1 + 1 * (1 + x + 0), x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
naiverev(z) -{ 1 }→ 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed:
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n1) [1 + z], size: O(n1) [z + z'] naiverev: runtime: O(n2) [4 + 3·z + 2·z2], size: O(n1) [z] goal: runtime: O(n2) [5 + 3·z + 2·z2], size: O(n1) [z]

(37) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(38) BOUNDS(1, n^2)