The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(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^3).


The TRS R consists of the following rules:

shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs))) [1]
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil)) [1]
append(Cons(x, xs), ys) → Cons(x, append(xs, ys)) [1]
shuffle(Nil) → Nil [1]
reverse(Nil) → Nil [1]
append(Nil, ys) → ys [1]
goal(xs) → shuffle(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:

shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs))) [1]
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil)) [1]
append(Cons(x, xs), ys) → Cons(x, append(xs, ys)) [1]
shuffle(Nil) → Nil [1]
reverse(Nil) → Nil [1]
append(Nil, ys) → ys [1]
goal(xs) → shuffle(xs) [1]

The TRS has the following type information:

shuffle :: Cons:Nil → Cons:Nil Cons :: a → Cons:Nil → Cons:Nil reverse :: Cons:Nil → Cons:Nil append :: Cons:Nil → Cons:Nil → Cons:Nil Nil :: Cons:Nil 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:

shuffle
goal

(c) The following functions are completely defined:

reverse
append

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: shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs))) [1] reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil)) [1] append(Cons(x, xs), ys) → Cons(x, append(xs, ys)) [1] shuffle(Nil) → Nil [1] reverse(Nil) → Nil [1] append(Nil, ys) → ys [1] goal(xs) → shuffle(xs) [1] The TRS has the following type information: shuffle :: Cons:Nil → Cons:Nil Cons :: a → Cons:Nil → Cons:Nil reverse :: Cons:Nil → Cons:Nil append :: Cons:Nil → Cons:Nil → Cons:Nil Nil :: Cons:Nil 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: shuffle(Cons(x, Cons(x', xs'))) → Cons(x, shuffle(append(reverse(xs'), Cons(x', Nil)))) [2] shuffle(Cons(x, Nil)) → Cons(x, shuffle(Nil)) [2] reverse(Cons(x, Cons(x'', xs''))) → append(append(reverse(xs''), Cons(x'', Nil)), Cons(x, Nil)) [2] reverse(Cons(x, Nil)) → append(Nil, Cons(x, Nil)) [2] append(Cons(x, xs), ys) → Cons(x, append(xs, ys)) [1] shuffle(Nil) → Nil [1] reverse(Nil) → Nil [1] append(Nil, ys) → ys [1] goal(xs) → shuffle(xs) [1] The TRS has the following type information: shuffle :: Cons:Nil → Cons:Nil Cons :: a → Cons:Nil → Cons:Nil reverse :: Cons:Nil → Cons:Nil append :: Cons:Nil → Cons:Nil → Cons:Nil Nil :: Cons:Nil 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
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

append(z, z') -{ 1 }→ ys :|: z' = ys, ys >= 0, z = 0
append(z, z') -{ 1 }→ 1 + x + append(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
goal(z) -{ 1 }→ shuffle(xs) :|: xs >= 0, z = xs
reverse(z) -{ 2 }→ append(append(reverse(xs''), 1 + x'' + 0), 1 + x + 0) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 2 }→ append(0, 1 + x + 0) :|: x >= 0, z = 1 + x + 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + x + shuffle(append(reverse(xs'), 1 + x' + 0)) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + x + shuffle(0) :|: x >= 0, z = 1 + x + 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:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(z, z') -{ 1 }→ 1 + x + append(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 2 }→ append(append(reverse(xs''), 1 + x'' + 0), 1 + x + 0) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 2 }→ append(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + x + shuffle(append(reverse(xs'), 1 + x' + 0)) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ append } { reverse } { shuffle } { goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(z, z') -{ 1 }→ 1 + x + append(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 2 }→ append(append(reverse(xs''), 1 + x'' + 0), 1 + x + 0) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 2 }→ append(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + x + shuffle(append(reverse(xs'), 1 + x' + 0)) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

Function symbols to be analyzed: {append}, {reverse}, {shuffle}, {goal}

(15) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(z, z') -{ 1 }→ 1 + x + append(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 2 }→ append(append(reverse(xs''), 1 + x'' + 0), 1 + x + 0) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 2 }→ append(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + x + shuffle(append(reverse(xs'), 1 + x' + 0)) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

Function symbols to be analyzed: {append}, {reverse}, {shuffle}, {goal}
Previous analysis results are:

append: runtime: ?, size: O(n1) [z + z']

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(z, z') -{ 1 }→ 1 + x + append(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
goal(z) -{ 1 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 2 }→ append(append(reverse(xs''), 1 + x'' + 0), 1 + x + 0) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 2 }→ append(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + x + shuffle(append(reverse(xs'), 1 + x' + 0)) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

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

(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:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(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 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
reverse(z) -{ 2 }→ append(append(reverse(xs''), 1 + x'' + 0), 1 + x + 0) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + x + shuffle(append(reverse(xs'), 1 + x' + 0)) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

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

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(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 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
reverse(z) -{ 2 }→ append(append(reverse(xs''), 1 + x'' + 0), 1 + x + 0) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + x + shuffle(append(reverse(xs'), 1 + x' + 0)) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

append: runtime: O(n1) [1 + z], size: O(n1) [z + z'] reverse: runtime: ?, size: O(n1) [z]

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(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 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
reverse(z) -{ 2 }→ append(append(reverse(xs''), 1 + x'' + 0), 1 + x + 0) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + x + shuffle(append(reverse(xs'), 1 + x' + 0)) :|: x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

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

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(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 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
reverse(z) -{ 8 + s2 + s3 + 3·xs'' + 2·xs''² }→ s4 :|: s2 >= 0, s2 <= 1 * xs'', s3 >= 0, s3 <= 1 * s2 + 1 * (1 + x'' + 0), s4 >= 0, s4 <= 1 * s3 + 1 * (1 + x + 0), z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 7 + s'' + 3·xs' + 2·xs'² }→ 1 + x + shuffle(s1) :|: s'' >= 0, s'' <= 1 * xs', s1 >= 0, s1 <= 1 * s'' + 1 * (1 + x' + 0), x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

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

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

Computed SIZE bound using KoAT for: shuffle
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:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(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 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
reverse(z) -{ 8 + s2 + s3 + 3·xs'' + 2·xs''² }→ s4 :|: s2 >= 0, s2 <= 1 * xs'', s3 >= 0, s3 <= 1 * s2 + 1 * (1 + x'' + 0), s4 >= 0, s4 <= 1 * s3 + 1 * (1 + x + 0), z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 7 + s'' + 3·xs' + 2·xs'² }→ 1 + x + shuffle(s1) :|: s'' >= 0, s'' <= 1 * xs', s1 >= 0, s1 <= 1 * s'' + 1 * (1 + x' + 0), x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

append: runtime: O(n1) [1 + z], size: O(n1) [z + z'] reverse: runtime: O(n2) [4 + 3·z + 2·z2], size: O(n1) [z] shuffle: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using KoAT for: shuffle
after applying outer abstraction to obtain an ITS,
resulting in: O(n³) with polynomial bound: 1 + 9·z + 4·z² + 2·z³

(30) Obligation:

Complexity RNTS consisting of the following rules:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(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 }→ shuffle(z) :|: z >= 0
reverse(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
reverse(z) -{ 8 + s2 + s3 + 3·xs'' + 2·xs''² }→ s4 :|: s2 >= 0, s2 <= 1 * xs'', s3 >= 0, s3 <= 1 * s2 + 1 * (1 + x'' + 0), s4 >= 0, s4 <= 1 * s3 + 1 * (1 + x + 0), z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 7 + s'' + 3·xs' + 2·xs'² }→ 1 + x + shuffle(s1) :|: s'' >= 0, s'' <= 1 * xs', s1 >= 0, s1 <= 1 * s'' + 1 * (1 + x' + 0), x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 2 }→ 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0

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

append: runtime: O(n1) [1 + z], size: O(n1) [z + z'] reverse: runtime: O(n2) [4 + 3·z + 2·z2], size: O(n1) [z] shuffle: runtime: O(n3) [1 + 9·z + 4·z2 + 2·z3], 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:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(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) -{ 2 + 9·z + 4·z² + 2·z³ }→ s7 :|: s7 >= 0, s7 <= 1 * z, z >= 0
reverse(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
reverse(z) -{ 8 + s2 + s3 + 3·xs'' + 2·xs''² }→ s4 :|: s2 >= 0, s2 <= 1 * xs'', s3 >= 0, s3 <= 1 * s2 + 1 * (1 + x'' + 0), s4 >= 0, s4 <= 1 * s3 + 1 * (1 + x + 0), z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 8 + s'' + 9·s1 + 4·s1² + 2·s1³ + 3·xs' + 2·xs'² }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * s1, s'' >= 0, s'' <= 1 * xs', s1 >= 0, s1 <= 1 * s'' + 1 * (1 + x' + 0), x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 3 }→ 1 + (z - 1) + s6 :|: s6 >= 0, s6 <= 1 * 0, z - 1 >= 0

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

append: runtime: O(n1) [1 + z], size: O(n1) [z + z'] reverse: runtime: O(n2) [4 + 3·z + 2·z2], size: O(n1) [z] shuffle: runtime: O(n3) [1 + 9·z + 4·z2 + 2·z3], 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:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(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) -{ 2 + 9·z + 4·z² + 2·z³ }→ s7 :|: s7 >= 0, s7 <= 1 * z, z >= 0
reverse(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
reverse(z) -{ 8 + s2 + s3 + 3·xs'' + 2·xs''² }→ s4 :|: s2 >= 0, s2 <= 1 * xs'', s3 >= 0, s3 <= 1 * s2 + 1 * (1 + x'' + 0), s4 >= 0, s4 <= 1 * s3 + 1 * (1 + x + 0), z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 8 + s'' + 9·s1 + 4·s1² + 2·s1³ + 3·xs' + 2·xs'² }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * s1, s'' >= 0, s'' <= 1 * xs', s1 >= 0, s1 <= 1 * s'' + 1 * (1 + x' + 0), x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 3 }→ 1 + (z - 1) + s6 :|: s6 >= 0, s6 <= 1 * 0, z - 1 >= 0

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

append: runtime: O(n1) [1 + z], size: O(n1) [z + z'] reverse: runtime: O(n2) [4 + 3·z + 2·z2], size: O(n1) [z] shuffle: runtime: O(n3) [1 + 9·z + 4·z2 + 2·z3], 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(n³) with polynomial bound: 2 + 9·z + 4·z² + 2·z³

(36) Obligation:

Complexity RNTS consisting of the following rules:

append(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
append(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) -{ 2 + 9·z + 4·z² + 2·z³ }→ s7 :|: s7 >= 0, s7 <= 1 * z, z >= 0
reverse(z) -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * (1 + (z - 1) + 0), z - 1 >= 0
reverse(z) -{ 8 + s2 + s3 + 3·xs'' + 2·xs''² }→ s4 :|: s2 >= 0, s2 <= 1 * xs'', s3 >= 0, s3 <= 1 * s2 + 1 * (1 + x'' + 0), s4 >= 0, s4 <= 1 * s3 + 1 * (1 + x + 0), z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 8 + s'' + 9·s1 + 4·s1² + 2·s1³ + 3·xs' + 2·xs'² }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * s1, s'' >= 0, s'' <= 1 * xs', s1 >= 0, s1 <= 1 * s'' + 1 * (1 + x' + 0), x >= 0, x' >= 0, xs' >= 0, z = 1 + x + (1 + x' + xs')
shuffle(z) -{ 3 }→ 1 + (z - 1) + s6 :|: s6 >= 0, s6 <= 1 * 0, z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

append: runtime: O(n1) [1 + z], size: O(n1) [z + z'] reverse: runtime: O(n2) [4 + 3·z + 2·z2], size: O(n1) [z] shuffle: runtime: O(n3) [1 + 9·z + 4·z2 + 2·z3], size: O(n1) [z] goal: runtime: O(n3) [2 + 9·z + 4·z2 + 2·z3], size: O(n1) [z]

(37) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(38) BOUNDS(1, n^3)