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:

merge(x, nil) → x
merge(nil, y) → y
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

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:

merge(x, nil) → x [1]
merge(nil, y) → y [1]
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v))) [1]
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v)) [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:

merge(x, nil) → x [1]
merge(nil, y) → y [1]
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v))) [1]
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v)) [1]

The TRS has the following type information:

merge :: nil:++:v → nil:++:v → nil:++:v nil :: nil:++:v ++ :: u → nil:++:v → nil:++:v u :: u v :: nil:++:v

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:

merge

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

(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: merge(x, nil) → x [1] merge(nil, y) → y [1] merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v))) [1] merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v)) [1] The TRS has the following type information: merge :: nil:++:v → nil:++:v → nil:++:v nil :: nil:++:v ++ :: u → nil:++:v → nil:++:v u :: u v :: nil:++:v 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: merge(x, nil) → x [1] merge(nil, y) → y [1] merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v))) [1] merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v)) [1] The TRS has the following type information: merge :: nil:++:v → nil:++:v → nil:++:v nil :: nil:++:v ++ :: u → nil:++:v → nil:++:v u :: u v :: nil:++:v 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
u => 0
v => 1

(10) Obligation:

Complexity RNTS consisting of the following rules:

merge(z, z') -{ 1 }→ x :|: x >= 0, z = x, z' = 0
merge(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
merge(z, z') -{ 1 }→ 1 + x + merge(y, 1 + 0 + 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 0
merge(z, z') -{ 1 }→ 1 + 0 + merge(1 + x + y, 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 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:

merge(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
merge(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
merge(z, z') -{ 1 }→ 1 + x + merge(y, 1 + 0 + 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 0
merge(z, z') -{ 1 }→ 1 + 0 + merge(1 + x + y, 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 0

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

Found the following analysis order by SCC decomposition:

{ merge }

(14) Obligation:

Complexity RNTS consisting of the following rules:

merge(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
merge(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
merge(z, z') -{ 1 }→ 1 + x + merge(y, 1 + 0 + 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 0
merge(z, z') -{ 1 }→ 1 + 0 + merge(1 + x + y, 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 0

Function symbols to be analyzed: {merge}

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

Computed SIZE bound using KoAT for: merge
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:

merge(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
merge(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
merge(z, z') -{ 1 }→ 1 + x + merge(y, 1 + 0 + 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 0
merge(z, z') -{ 1 }→ 1 + 0 + merge(1 + x + y, 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 0

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

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

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

Computed RUNTIME bound using CoFloCo for: merge
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2 + z

(18) Obligation:

Complexity RNTS consisting of the following rules:

merge(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
merge(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
merge(z, z') -{ 1 }→ 1 + x + merge(y, 1 + 0 + 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 0
merge(z, z') -{ 1 }→ 1 + 0 + merge(1 + x + y, 1) :|: z = 1 + x + y, z' = 1 + 0 + 1, x >= 0, y >= 0

Function symbols to be analyzed:
Previous analysis results are:

merge: runtime: O(n1) [2 + z], size: O(n1) [z + z']

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^1)