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

(0) Obligation:

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


The TRS R consists of the following rules:

append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
append(Nil, ys) → ys
goal(x, y) → append(x, y)

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^1).


The TRS R consists of the following rules:

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

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

The TRS has the following type information:

append :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: a → Cons:Nil → Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil → 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:

append
goal

(c) The following functions are completely defined:
none

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: append(Cons(x, xs), ys) → Cons(x, append(xs, ys)) [1] append(Nil, ys) → ys [1] goal(x, y) → append(x, y) [1] The TRS has the following type information: append :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: a → Cons:Nil → Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil → 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: append(Cons(x, xs), ys) → Cons(x, append(xs, ys)) [1] append(Nil, ys) → ys [1] goal(x, y) → append(x, y) [1] The TRS has the following type information: append :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: a → Cons:Nil → Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil → 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, z') -{ 1 }→ append(x, y) :|: x >= 0, y >= 0, z = x, z' = y

(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, z') -{ 1 }→ append(z, z') :|: z >= 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ append } { 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, z') -{ 1 }→ append(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {append}, {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, z') -{ 1 }→ append(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {append}, {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, z') -{ 1 }→ append(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {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, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z', z >= 0, z' >= 0

Function symbols to be analyzed: {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: goal
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:

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, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z', z >= 0, z' >= 0

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

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

(23) 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: 2 + z

(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, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z', z >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

append: runtime: O(n1) [1 + z], size: O(n1) [z + z'] goal: runtime: O(n1) [2 + z], size: O(n1) [z + z']

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^1)