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:

app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))

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:

app(nil, y) → y [1]
app(add(n, x), y) → add(n, app(x, y)) [1]
reverse(nil) → nil [1]
reverse(add(n, x)) → app(reverse(x), add(n, nil)) [1]
shuffle(nil) → nil [1]
shuffle(add(n, x)) → add(n, shuffle(reverse(x))) [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:

app(nil, y) → y [1]
app(add(n, x), y) → add(n, app(x, y)) [1]
reverse(nil) → nil [1]
reverse(add(n, x)) → app(reverse(x), add(n, nil)) [1]
shuffle(nil) → nil [1]
shuffle(add(n, x)) → add(n, shuffle(reverse(x))) [1]

The TRS has the following type information:

app :: nil:add → nil:add → nil:add nil :: nil:add add :: a → nil:add → nil:add reverse :: nil:add → nil:add shuffle :: nil:add → nil:add

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

(c) The following functions are completely defined:

reverse
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: app(nil, y) → y [1] app(add(n, x), y) → add(n, app(x, y)) [1] reverse(nil) → nil [1] reverse(add(n, x)) → app(reverse(x), add(n, nil)) [1] shuffle(nil) → nil [1] shuffle(add(n, x)) → add(n, shuffle(reverse(x))) [1] The TRS has the following type information: app :: nil:add → nil:add → nil:add nil :: nil:add add :: a → nil:add → nil:add reverse :: nil:add → nil:add shuffle :: nil:add → nil:add 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: app(nil, y) → y [1] app(add(n, x), y) → add(n, app(x, y)) [1] reverse(nil) → nil [1] reverse(add(n, nil)) → app(nil, add(n, nil)) [2] reverse(add(n, add(n', x'))) → app(app(reverse(x'), add(n', nil)), add(n, nil)) [2] shuffle(nil) → nil [1] shuffle(add(n, nil)) → add(n, shuffle(nil)) [2] shuffle(add(n, add(n'', x''))) → add(n, shuffle(app(reverse(x''), add(n'', nil)))) [2] The TRS has the following type information: app :: nil:add → nil:add → nil:add nil :: nil:add add :: a → nil:add → nil:add reverse :: nil:add → nil:add shuffle :: nil:add → nil:add 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:

app(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
app(z, z') -{ 1 }→ 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y
reverse(z) -{ 2 }→ app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
reverse(z) -{ 2 }→ app(0, 1 + n + 0) :|: z = 1 + n + 0, n >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
shuffle(z) -{ 2 }→ 1 + n + shuffle(0) :|: z = 1 + n + 0, n >= 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 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
reverse(z) -{ 2 }→ app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0
reverse(z) -{ 2 }→ app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
reverse(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 1 }→ 0 :|: z = 0
shuffle(z) -{ 2 }→ 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'')
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:

{ app } { reverse } { shuffle }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {app}, {reverse}, {shuffle}

(15) 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'

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

app: 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:

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

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

app: 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:

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

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

app: 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:

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

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

app: 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:

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

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

app: 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:

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

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

app: 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:

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

Function symbols to be analyzed:
Previous analysis results are:

app: 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) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^3)