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:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

Rewrite Strategy: INNERMOST

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 1th argument of plus: times

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))

(2) 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:

times(X, s(Y)) → plus(X, times(Y, X))

Rewrite Strategy: INNERMOST

(3) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(4) 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:

times(X, s(Y)) → plus(X, times(Y, X)) [1]

Rewrite Strategy: INNERMOST

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

times(X, s(Y)) → plus(X, times(Y, X)) [1]

The TRS has the following type information:

times :: s → s → plus s :: s → s plus :: s → plus → plus

Rewrite Strategy: INNERMOST

(7) 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:

times

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const, const1

(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: times(X, s(Y)) → plus(X, times(Y, X)) [1] The TRS has the following type information: times :: s → s → plus s :: s → s plus :: s → plus → plus const :: plus const1 :: s Rewrite Strategy: INNERMOST

(9) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(10) 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: times(X, s(Y)) → plus(X, times(Y, X)) [1] The TRS has the following type information: times :: s → s → plus s :: s → s plus :: s → plus → plus const :: plus const1 :: s Rewrite Strategy: INNERMOST

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

const => 0
const1 => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

times(z, z') -{ 1 }→ 1 + X + times(Y, X) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X

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

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

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

Found the following analysis order by SCC decomposition:

{ times }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {times}

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

Computed SIZE bound using CoFloCo for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

times: runtime: ?, size: O(1) [0]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

times: runtime: O(n1) [z + z'], size: O(1) [0]

(21) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(22) BOUNDS(1, n^1)