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:

f(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

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:

f(empty, l) → l [1]
f(cons(x, k), l) → g(k, l, cons(x, k)) [1]
g(a, b, c) → f(a, cons(b, c)) [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:

f(empty, l) → l [1]
f(cons(x, k), l) → g(k, l, cons(x, k)) [1]
g(a, b, c) → f(a, cons(b, c)) [1]

The TRS has the following type information:

f :: empty:cons → empty:cons → empty:cons empty :: empty:cons cons :: empty:cons → empty:cons → empty:cons g :: empty:cons → empty:cons → empty:cons → empty:cons

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:

f
g

(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: f(empty, l) → l [1] f(cons(x, k), l) → g(k, l, cons(x, k)) [1] g(a, b, c) → f(a, cons(b, c)) [1] The TRS has the following type information: f :: empty:cons → empty:cons → empty:cons empty :: empty:cons cons :: empty:cons → empty:cons → empty:cons g :: empty:cons → empty:cons → empty:cons → empty:cons 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: f(empty, l) → l [1] f(cons(x, k), l) → g(k, l, cons(x, k)) [1] g(a, b, c) → f(a, cons(b, c)) [1] The TRS has the following type information: f :: empty:cons → empty:cons → empty:cons empty :: empty:cons cons :: empty:cons → empty:cons → empty:cons g :: empty:cons → empty:cons → empty:cons → empty:cons 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:

empty => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 1 }→ l :|: z' = l, l >= 0, z = 0
f(z, z') -{ 1 }→ g(k, l, 1 + x + k) :|: z' = l, x >= 0, l >= 0, k >= 0, z = 1 + x + k
g(z, z', z'') -{ 1 }→ f(a, 1 + b + c) :|: z = a, b >= 0, a >= 0, c >= 0, z' = b, z'' = c

(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:

f(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z, z') -{ 1 }→ g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k
g(z, z', z'') -{ 1 }→ f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0

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

Found the following analysis order by SCC decomposition:

{ f, g }

(14) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z, z') -{ 1 }→ g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k
g(z, z', z'') -{ 1 }→ f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0

Function symbols to be analyzed: {f,g}

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

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: z + z² + z'

Computed SIZE bound using KoAT for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + z + z² + z' + z''

(16) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z, z') -{ 1 }→ g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k
g(z, z', z'') -{ 1 }→ f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0

Function symbols to be analyzed: {f,g}
Previous analysis results are:

f: runtime: ?, size: O(n2) [z + z2 + z'] g: runtime: ?, size: O(n2) [1 + z + z2 + z' + z'']

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

Computed RUNTIME bound using PUBS for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 2·z

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z, z') -{ 1 }→ g(k, z', 1 + x + k) :|: x >= 0, z' >= 0, k >= 0, z = 1 + x + k
g(z, z', z'') -{ 1 }→ f(z, 1 + z' + z'') :|: z' >= 0, z >= 0, z'' >= 0

Function symbols to be analyzed:
Previous analysis results are:

f: runtime: O(n1) [1 + 2·z], size: O(n2) [z + z2 + z'] g: runtime: O(n1) [2 + 2·z], size: O(n2) [1 + z + z2 + z' + z'']

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^1)