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:

a__f(f(a)) → a__f(g(f(a)))
mark(f(X)) → a__f(X)
mark(a) → a
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

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:

a__f(f(a)) → a__f(g(f(a))) [1]
mark(f(X)) → a__f(X) [1]
mark(a) → a [1]
mark(g(X)) → g(mark(X)) [1]
a__f(X) → f(X) [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:

a__f(f(a)) → a__f(g(f(a))) [1]
mark(f(X)) → a__f(X) [1]
mark(a) → a [1]
mark(g(X)) → g(mark(X)) [1]
a__f(X) → f(X) [1]

The TRS has the following type information:

a__f :: a:f:g → a:f:g f :: a:f:g → a:f:g a :: a:f:g g :: a:f:g → a:f:g mark :: a:f:g → a:f:g

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:

a__f
mark

(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: a__f(f(a)) → a__f(g(f(a))) [1] mark(f(X)) → a__f(X) [1] mark(a) → a [1] mark(g(X)) → g(mark(X)) [1] a__f(X) → f(X) [1] The TRS has the following type information: a__f :: a:f:g → a:f:g f :: a:f:g → a:f:g a :: a:f:g g :: a:f:g → a:f:g mark :: a:f:g → a:f:g 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: a__f(f(a)) → a__f(g(f(a))) [1] mark(f(X)) → a__f(X) [1] mark(a) → a [1] mark(g(X)) → g(mark(X)) [1] a__f(X) → f(X) [1] The TRS has the following type information: a__f :: a:f:g → a:f:g f :: a:f:g → a:f:g a :: a:f:g g :: a:f:g → a:f:g mark :: a:f:g → a:f:g 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:

a => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 1 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
mark(z) -{ 1 }→ a__f(X) :|: z = 1 + X, X >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + mark(X) :|: z = 1 + X, X >= 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:

a__f(z) -{ 1 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 1 }→ a__f(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ a__f } { mark }

(14) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 1 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 1 }→ a__f(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {a__f}, {mark}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 1 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 1 }→ a__f(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:

a__f: runtime: ?, size: O(n1) [2 + z]

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

Computed RUNTIME bound using CoFloCo for: a__f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(18) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 1 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 1 }→ a__f(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0

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

a__f: runtime: O(1) [2], size: O(n1) [2 + 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:

a__f(z) -{ 3 }→ s :|: s >= 0, s <= 1 * (1 + (1 + 0)) + 2, z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 3 }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 2, z - 1 >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0

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

a__f: runtime: O(1) [2], size: O(n1) [2 + z]

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z

(22) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 3 }→ s :|: s >= 0, s <= 1 * (1 + (1 + 0)) + 2, z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 3 }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 2, z - 1 >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0

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

a__f: runtime: O(1) [2], size: O(n1) [2 + z] mark: runtime: ?, size: O(n1) [1 + z]

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

Computed RUNTIME bound using PUBS for: mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + z

(24) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 3 }→ s :|: s >= 0, s <= 1 * (1 + (1 + 0)) + 2, z = 1 + 0
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 3 }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 2, z - 1 >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

a__f: runtime: O(1) [2], size: O(n1) [2 + z] mark: runtime: O(n1) [3 + z], size: O(n1) [1 + z]

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^1)