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(X)) → a__c(f(g(f(X))))
a__c(X) → d(X)
a__h(X) → a__c(d(X))
mark(f(X)) → a__f(mark(X))
mark(c(X)) → a__c(X)
mark(h(X)) → a__h(mark(X))
mark(g(X)) → g(X)
mark(d(X)) → d(X)
a__f(X) → f(X)
a__c(X) → c(X)
a__h(X) → h(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(X)) → a__c(f(g(f(X)))) [1]
a__c(X) → d(X) [1]
a__h(X) → a__c(d(X)) [1]
mark(f(X)) → a__f(mark(X)) [1]
mark(c(X)) → a__c(X) [1]
mark(h(X)) → a__h(mark(X)) [1]
mark(g(X)) → g(X) [1]
mark(d(X)) → d(X) [1]
a__f(X) → f(X) [1]
a__c(X) → c(X) [1]
a__h(X) → h(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(X)) → a__c(f(g(f(X)))) [1]
a__c(X) → d(X) [1]
a__h(X) → a__c(d(X)) [1]
mark(f(X)) → a__f(mark(X)) [1]
mark(c(X)) → a__c(X) [1]
mark(h(X)) → a__h(mark(X)) [1]
mark(g(X)) → g(X) [1]
mark(d(X)) → d(X) [1]
a__f(X) → f(X) [1]
a__c(X) → c(X) [1]
a__h(X) → h(X) [1]

The TRS has the following type information:

a__f :: f:g:d:c:h → f:g:d:c:h f :: f:g:d:c:h → f:g:d:c:h a__c :: f:g:d:c:h → f:g:d:c:h g :: f:g:d:c:h → f:g:d:c:h d :: f:g:d:c:h → f:g:d:c:h a__h :: f:g:d:c:h → f:g:d:c:h mark :: f:g:d:c:h → f:g:d:c:h c :: f:g:d:c:h → f:g:d:c:h h :: f:g:d:c:h → f:g:d:c:h

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__h
a__c
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(f(X)) → a__c(f(g(f(X)))) [1] a__c(X) → d(X) [1] a__h(X) → a__c(d(X)) [1] mark(f(X)) → a__f(mark(X)) [1] mark(c(X)) → a__c(X) [1] mark(h(X)) → a__h(mark(X)) [1] mark(g(X)) → g(X) [1] mark(d(X)) → d(X) [1] a__f(X) → f(X) [1] a__c(X) → c(X) [1] a__h(X) → h(X) [1] mark(v0) → const [0] The TRS has the following type information: a__f :: f:g:d:c:h:const → f:g:d:c:h:const f :: f:g:d:c:h:const → f:g:d:c:h:const a__c :: f:g:d:c:h:const → f:g:d:c:h:const g :: f:g:d:c:h:const → f:g:d:c:h:const d :: f:g:d:c:h:const → f:g:d:c:h:const a__h :: f:g:d:c:h:const → f:g:d:c:h:const mark :: f:g:d:c:h:const → f:g:d:c:h:const c :: f:g:d:c:h:const → f:g:d:c:h:const h :: f:g:d:c:h:const → f:g:d:c:h:const const :: f:g:d:c:h: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(f(X)) → a__c(f(g(f(X)))) [1] a__c(X) → d(X) [1] a__h(X) → a__c(d(X)) [1] mark(f(f(X'))) → a__f(a__f(mark(X'))) [2] mark(f(c(X''))) → a__f(a__c(X'')) [2] mark(f(h(X1))) → a__f(a__h(mark(X1))) [2] mark(f(g(X2))) → a__f(g(X2)) [2] mark(f(d(X3))) → a__f(d(X3)) [2] mark(f(X)) → a__f(const) [1] mark(c(X)) → a__c(X) [1] mark(h(f(X4))) → a__h(a__f(mark(X4))) [2] mark(h(c(X5))) → a__h(a__c(X5)) [2] mark(h(h(X6))) → a__h(a__h(mark(X6))) [2] mark(h(g(X7))) → a__h(g(X7)) [2] mark(h(d(X8))) → a__h(d(X8)) [2] mark(h(X)) → a__h(const) [1] mark(g(X)) → g(X) [1] mark(d(X)) → d(X) [1] a__f(X) → f(X) [1] a__c(X) → c(X) [1] a__h(X) → h(X) [1] mark(v0) → const [0] The TRS has the following type information: a__f :: f:g:d:c:h:const → f:g:d:c:h:const f :: f:g:d:c:h:const → f:g:d:c:h:const a__c :: f:g:d:c:h:const → f:g:d:c:h:const g :: f:g:d:c:h:const → f:g:d:c:h:const d :: f:g:d:c:h:const → f:g:d:c:h:const a__h :: f:g:d:c:h:const → f:g:d:c:h:const mark :: f:g:d:c:h:const → f:g:d:c:h:const c :: f:g:d:c:h:const → f:g:d:c:h:const h :: f:g:d:c:h:const → f:g:d:c:h:const const :: f:g:d:c:h: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__c(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a__f(z) -{ 1 }→ a__c(1 + (1 + (1 + X))) :|: z = 1 + X, X >= 0
a__f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a__h(z) -{ 1 }→ a__c(1 + X) :|: X >= 0, z = X
a__h(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
mark(z) -{ 2 }→ a__h(a__h(mark(X6))) :|: X6 >= 0, z = 1 + (1 + X6)
mark(z) -{ 2 }→ a__h(a__f(mark(X4))) :|: z = 1 + (1 + X4), X4 >= 0
mark(z) -{ 2 }→ a__h(a__c(X5)) :|: X5 >= 0, z = 1 + (1 + X5)
mark(z) -{ 1 }→ a__h(0) :|: z = 1 + X, X >= 0
mark(z) -{ 2 }→ a__h(1 + X7) :|: X7 >= 0, z = 1 + (1 + X7)
mark(z) -{ 2 }→ a__h(1 + X8) :|: X8 >= 0, z = 1 + (1 + X8)
mark(z) -{ 2 }→ a__f(a__h(mark(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
mark(z) -{ 2 }→ a__f(a__f(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
mark(z) -{ 2 }→ a__f(a__c(X'')) :|: z = 1 + (1 + X''), X'' >= 0
mark(z) -{ 1 }→ a__f(0) :|: z = 1 + X, X >= 0
mark(z) -{ 2 }→ a__f(1 + X2) :|: z = 1 + (1 + X2), X2 >= 0
mark(z) -{ 2 }→ a__f(1 + X3) :|: z = 1 + (1 + X3), X3 >= 0
mark(z) -{ 1 }→ a__c(X) :|: z = 1 + X, X >= 0
mark(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
mark(z) -{ 1 }→ 1 + X :|: z = 1 + X, X >= 0

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

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

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a__f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a__f(z) -{ 2 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, 1 + (1 + (1 + X)) = X'
a__h(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a__h(z) -{ 2 }→ 1 + X' :|: X >= 0, z = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(X6))) :|: X6 >= 0, z = 1 + (1 + X6)
mark(z) -{ 2 }→ a__h(a__f(mark(X4))) :|: z = 1 + (1 + X4), X4 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
mark(z) -{ 2 }→ a__f(a__f(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
mark(z) -{ 3 }→ a__c(1 + X) :|: X7 >= 0, z = 1 + (1 + X7), X >= 0, 1 + X7 = X
mark(z) -{ 3 }→ a__c(1 + X) :|: X8 >= 0, z = 1 + (1 + X8), X >= 0, 1 + X8 = X
mark(z) -{ 2 }→ a__c(1 + X') :|: z = 1 + X, X >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ a__c(1 + X') :|: X5 >= 0, z = 1 + (1 + X5), X >= 0, X5 = X, X' >= 0, 1 + X = X'
mark(z) -{ 3 }→ a__c(1 + (1 + (1 + X))) :|: z = 1 + (1 + X2), X2 >= 0, 1 + X2 = 1 + X, X >= 0
mark(z) -{ 3 }→ a__c(1 + (1 + (1 + X))) :|: z = 1 + (1 + X3), X3 >= 0, 1 + X3 = 1 + X, X >= 0
mark(z) -{ 4 }→ a__c(1 + (1 + (1 + X'))) :|: z = 1 + (1 + X''), X'' >= 0, X >= 0, X'' = X, 1 + X = 1 + X', X' >= 0
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 + X2), X2 >= 0, X >= 0, 1 + X2 = X
mark(z) -{ 3 }→ 1 + X :|: z = 1 + (1 + X3), X3 >= 0, X >= 0, 1 + X3 = X
mark(z) -{ 3 }→ 1 + X :|: X7 >= 0, z = 1 + (1 + X7), X >= 0, 1 + X7 = X
mark(z) -{ 3 }→ 1 + X :|: X8 >= 0, z = 1 + (1 + X8), X >= 0, 1 + X8 = X
mark(z) -{ 2 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
mark(z) -{ 2 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z = 1 + (1 + X''), X'' >= 0, X >= 0, X'' = X, X' >= 0, 1 + X = X'
mark(z) -{ 4 }→ 1 + X' :|: X5 >= 0, z = 1 + (1 + X5), X >= 0, X5 = X, X' >= 0, 1 + X = 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__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__c(1 + X) :|: z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 2 }→ a__c(1 + X') :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ a__c(1 + X') :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 3 }→ a__c(1 + (1 + (1 + X))) :|: z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 4 }→ a__c(1 + (1 + (1 + X'))) :|: z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ a__c } { a__h } { a__f } { mark }

(16) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__c(1 + X) :|: z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 2 }→ a__c(1 + X') :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ a__c(1 + X') :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 3 }→ a__c(1 + (1 + (1 + X))) :|: z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 4 }→ a__c(1 + (1 + (1 + X'))) :|: z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__c(1 + X) :|: z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 2 }→ a__c(1 + X') :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ a__c(1 + X') :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 3 }→ a__c(1 + (1 + (1 + X))) :|: z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 4 }→ a__c(1 + (1 + (1 + X'))) :|: z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

a__c: runtime: ?, size: O(n1) [1 + z]

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

Computed RUNTIME bound using CoFloCo for: a__c
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__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__c(1 + X) :|: z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 2 }→ a__c(1 + X') :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ a__c(1 + X') :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 3 }→ a__c(1 + (1 + (1 + X))) :|: z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 4 }→ a__c(1 + (1 + (1 + X'))) :|: z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

a__c: runtime: O(1) [1], size: O(n1) [1 + 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__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * (1 + (1 + (1 + X))) + 1, z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + (1 + X'))) + 1, z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 0
mark(z) -{ 4 }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + X) + 1, z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + X') + 1, z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 5 }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + X') + 1, z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * (1 + (1 + (1 + X))) + 1, z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + (1 + X'))) + 1, z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 0
mark(z) -{ 4 }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + X) + 1, z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + X') + 1, z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 5 }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + X') + 1, z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

a__c: runtime: O(1) [1], size: O(n1) [1 + z] a__h: runtime: ?, size: O(n1) [2 + z]

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

Computed RUNTIME bound using KoAT for: a__h
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

(26) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * (1 + (1 + (1 + X))) + 1, z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + (1 + X'))) + 1, z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 0
mark(z) -{ 4 }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + X) + 1, z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + X') + 1, z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 5 }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + X') + 1, z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

a__c: runtime: O(1) [1], size: O(n1) [1 + z] a__h: runtime: O(1) [3], size: O(n1) [2 + z]

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * (1 + (1 + (1 + X))) + 1, z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + (1 + X'))) + 1, z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 0
mark(z) -{ 4 }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + X) + 1, z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + X') + 1, z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 5 }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + X') + 1, z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

a__c: runtime: O(1) [1], size: O(n1) [1 + z] a__h: runtime: O(1) [3], size: O(n1) [2 + z]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * (1 + (1 + (1 + X))) + 1, z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + (1 + X'))) + 1, z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 0
mark(z) -{ 4 }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + X) + 1, z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + X') + 1, z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 5 }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + X') + 1, z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: a__f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

(32) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * (1 + (1 + (1 + X))) + 1, z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + (1 + X'))) + 1, z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 0
mark(z) -{ 4 }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + X) + 1, z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + X') + 1, z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 5 }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + X') + 1, z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

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

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * (1 + (1 + (1 + X))) + 1, z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + (1 + X'))) + 1, z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 0
mark(z) -{ 4 }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + X) + 1, z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + X') + 1, z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 5 }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + X') + 1, z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

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

(35) 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

(36) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * (1 + (1 + (1 + X))) + 1, z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + (1 + X'))) + 1, z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 0
mark(z) -{ 4 }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + X) + 1, z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + X') + 1, z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 5 }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + X') + 1, z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

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

(37) 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: 33 + 8·z

(38) Obligation:

Complexity RNTS consisting of the following rules:

a__c(z) -{ 1 }→ 1 + z :|: z >= 0
a__f(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 1 + (1 + (1 + (z - 1))) = X'
a__f(z) -{ 1 }→ 1 + z :|: z >= 0
a__h(z) -{ 2 }→ 1 + X' :|: z >= 0, X' >= 0, 1 + z = X'
a__h(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * (1 + (1 + (1 + X))) + 1, z - 2 >= 0, 1 + (z - 2) = 1 + X, X >= 0
mark(z) -{ 5 }→ s' :|: s' >= 0, s' <= 1 * (1 + (1 + (1 + X'))) + 1, z - 2 >= 0, X >= 0, z - 2 = X, 1 + X = 1 + X', X' >= 0
mark(z) -{ 4 }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + X) + 1, z - 2 >= 0, X >= 0, 1 + (z - 2) = X
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + X') + 1, z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 5 }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + X') + 1, z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 2 }→ a__h(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__h(a__f(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__h(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(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, z - 1 = X'
mark(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, 0 = X'
mark(z) -{ 4 }→ 1 + X' :|: z - 2 >= 0, X >= 0, z - 2 = X, X' >= 0, 1 + X = X'
mark(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

a__c: runtime: O(1) [1], size: O(n1) [1 + z] a__h: runtime: O(1) [3], size: O(n1) [2 + z] a__f: runtime: O(1) [3], size: O(n1) [3 + z] mark: runtime: O(n1) [33 + 8·z], size: O(n1) [3·z]

(39) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(40) BOUNDS(1, n^1)