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:

w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

Rewrite Strategy: INNERMOST

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
b(w(x)) → w(b(x))

(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:

b(r(x)) → r(b(x))
w(r(x)) → r(w(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:

b(r(x)) → r(b(x)) [1]
w(r(x)) → r(w(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:

b(r(x)) → r(b(x)) [1]
w(r(x)) → r(w(x)) [1]

The TRS has the following type information:

b :: r → r r :: r → r w :: r → r

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:

b
w

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const

(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: b(r(x)) → r(b(x)) [1] w(r(x)) → r(w(x)) [1] The TRS has the following type information: b :: r → r r :: r → r w :: r → r const :: r 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: b(r(x)) → r(b(x)) [1] w(r(x)) → r(w(x)) [1] The TRS has the following type information: b :: r → r r :: r → r w :: r → r const :: r 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

(12) Obligation:

Complexity RNTS consisting of the following rules:

b(z) -{ 1 }→ 1 + b(x) :|: x >= 0, z = 1 + x
w(z) -{ 1 }→ 1 + w(x) :|: x >= 0, z = 1 + 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:

b(z) -{ 1 }→ 1 + b(z - 1) :|: z - 1 >= 0
w(z) -{ 1 }→ 1 + w(z - 1) :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ w } { b }

(16) Obligation:

Complexity RNTS consisting of the following rules:

b(z) -{ 1 }→ 1 + b(z - 1) :|: z - 1 >= 0
w(z) -{ 1 }→ 1 + w(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {w}, {b}

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

Computed SIZE bound using CoFloCo for: w
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:

b(z) -{ 1 }→ 1 + b(z - 1) :|: z - 1 >= 0
w(z) -{ 1 }→ 1 + w(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {w}, {b}
Previous analysis results are:

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

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

b(z) -{ 1 }→ 1 + b(z - 1) :|: z - 1 >= 0
w(z) -{ 1 }→ 1 + w(z - 1) :|: z - 1 >= 0

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

w: runtime: O(n1) [z], size: O(1) [0]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(22) Obligation:

Complexity RNTS consisting of the following rules:

b(z) -{ 1 }→ 1 + b(z - 1) :|: z - 1 >= 0
w(z) -{ z }→ 1 + s :|: s >= 0, s <= 0, z - 1 >= 0

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

w: runtime: O(n1) [z], size: O(1) [0]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

b(z) -{ 1 }→ 1 + b(z - 1) :|: z - 1 >= 0
w(z) -{ z }→ 1 + s :|: s >= 0, s <= 0, z - 1 >= 0

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

w: runtime: O(n1) [z], size: O(1) [0] b: runtime: ?, size: O(1) [0]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

b(z) -{ 1 }→ 1 + b(z - 1) :|: z - 1 >= 0
w(z) -{ z }→ 1 + s :|: s >= 0, s <= 0, z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

w: runtime: O(n1) [z], size: O(1) [0] b: runtime: O(n1) [z], size: O(1) [0]

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, n^1)