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

The TRS has the following type information:

a__f :: f:h:g → f:h:g g :: f:h:g → f:h:g h :: f:h:g → f:h:g f :: f:h:g → f:h:g mark :: f:h:g → f:h: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:
none

(c) The following functions are completely defined:

mark
a__f

Due to the following rules being added:

mark(v0) → const [0]

And the following fresh constants:

const

(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(X) → g(h(f(X))) [1] mark(f(X)) → a__f(mark(X)) [1] mark(g(X)) → g(X) [1] mark(h(X)) → h(mark(X)) [1] a__f(X) → f(X) [1] mark(v0) → const [0] The TRS has the following type information: a__f :: f:h:g:const → f:h:g:const g :: f:h:g:const → f:h:g:const h :: f:h:g:const → f:h:g:const f :: f:h:g:const → f:h:g:const mark :: f:h:g:const → f:h:g:const const :: f:h:g:const 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(X) → g(h(f(X))) [1] mark(f(f(X'))) → a__f(a__f(mark(X'))) [2] mark(f(g(X''))) → a__f(g(X'')) [2] mark(f(h(X1))) → a__f(h(mark(X1))) [2] mark(f(X)) → a__f(const) [1] mark(g(X)) → g(X) [1] mark(h(X)) → h(mark(X)) [1] a__f(X) → f(X) [1] mark(v0) → const [0] The TRS has the following type information: a__f :: f:h:g:const → f:h:g:const g :: f:h:g:const → f:h:g:const h :: f:h:g:const → f:h:g:const f :: f:h:g:const → f:h:g:const mark :: f:h:g:const → f:h:g:const const :: f:h:g:const 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:

const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a__f(z) -{ 1 }→ 1 + (1 + (1 + X)) :|: X >= 0, z = X
mark(z) -{ 2 }→ a__f(a__f(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
mark(z) -{ 1 }→ a__f(0) :|: z = 1 + X, X >= 0
mark(z) -{ 2 }→ a__f(1 + X'') :|: z = 1 + (1 + X''), X'' >= 0
mark(z) -{ 2 }→ a__f(1 + mark(X1)) :|: X1 >= 0, z = 1 + (1 + X1)
mark(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
mark(z) -{ 1 }→ 1 + X :|: z = 1 + X, X >= 0
mark(z) -{ 1 }→ 1 + mark(X) :|: z = 1 + X, X >= 0

(11) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a__f(z) -{ 1 }→ 1 + (1 + (1 + X)) :|: X >= 0, z = X
mark(z) -{ 2 }→ a__f(a__f(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
mark(z) -{ 2 }→ a__f(1 + mark(X1)) :|: X1 >= 0, z = 1 + (1 + X1)
mark(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
mark(z) -{ 1 }→ 1 + X :|: z = 1 + X, X >= 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + (1 + X''), X'' >= 0, X >= 0, 1 + X'' = X
mark(z) -{ 2 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, 0 = X'
mark(z) -{ 1 }→ 1 + mark(X) :|: z = 1 + X, X >= 0
mark(z) -{ 3 }→ 1 + (1 + (1 + X)) :|: z = 1 + (1 + X''), X'' >= 0, X >= 0, 1 + X'' = X
mark(z) -{ 2 }→ 1 + (1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, 0 = 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:

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

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

Found the following analysis order by SCC decomposition:

{ a__f } { mark }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

a__f: runtime: ?, size: O(n1) [3 + z]

(19) 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: 1

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

(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:

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

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

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

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

Computed SIZE bound using KoAT 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) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(1 + mark(z - 2)) :|: z - 2 >= 0
mark(z) -{ 0 }→ 0 :|: z >= 0
mark(z) -{ 3 }→ 1 + X :|: z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 1 }→ 1 + mark(z - 1) :|: z - 1 >= 0
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0
mark(z) -{ 3 }→ 1 + (1 + (1 + X)) :|: z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 2 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, 0 = X'

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

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

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

Computed RUNTIME bound using KoAT for: mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 6 + 8·z

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

a__f: runtime: O(1) [1], size: O(n1) [3 + z] mark: runtime: O(n1) [6 + 8·z], size: O(n1) [3·z]

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, n^1)