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(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__b → b

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(X), Y) → a__f(Y, Y, Y) [1]
a__g(b) → c [1]
a__b → c [1]
mark(f(X1, X2, X3)) → a__f(X1, X2, X3) [1]
mark(g(X)) → a__g(mark(X)) [1]
mark(b) → a__b [1]
mark(c) → c [1]
a__f(X1, X2, X3) → f(X1, X2, X3) [1]
a__g(X) → g(X) [1]
a__b → b [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(X), Y) → a__f(Y, Y, Y) [1]
a__g(b) → c [1]
a__b → c [1]
mark(f(X1, X2, X3)) → a__f(X1, X2, X3) [1]
mark(g(X)) → a__g(mark(X)) [1]
mark(b) → a__b [1]
mark(c) → c [1]
a__f(X1, X2, X3) → f(X1, X2, X3) [1]
a__g(X) → g(X) [1]
a__b → b [1]

The TRS has the following type information:

a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f g :: g:b:c:f → g:b:c:f a__g :: g:b:c:f → g:b:c:f b :: g:b:c:f c :: g:b:c:f a__b :: g:b:c:f mark :: g:b:c:f → g:b:c:f f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f

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__b
a__f
a__g

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(X, g(X), Y) → a__f(Y, Y, Y) [1] a__g(b) → c [1] a__b → c [1] mark(f(X1, X2, X3)) → a__f(X1, X2, X3) [1] mark(g(X)) → a__g(mark(X)) [1] mark(b) → a__b [1] mark(c) → c [1] a__f(X1, X2, X3) → f(X1, X2, X3) [1] a__g(X) → g(X) [1] a__b → b [1] The TRS has the following type information: a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f g :: g:b:c:f → g:b:c:f a__g :: g:b:c:f → g:b:c:f b :: g:b:c:f c :: g:b:c:f a__b :: g:b:c:f mark :: g:b:c:f → g:b:c:f f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f 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(X), Y) → a__f(Y, Y, Y) [1] a__g(b) → c [1] a__b → c [1] mark(f(X1, X2, X3)) → a__f(X1, X2, X3) [1] mark(g(f(X1', X2', X3'))) → a__g(a__f(X1', X2', X3')) [2] mark(g(g(X'))) → a__g(a__g(mark(X'))) [2] mark(g(b)) → a__g(a__b) [2] mark(g(c)) → a__g(c) [2] mark(b) → a__b [1] mark(c) → c [1] a__f(X1, X2, X3) → f(X1, X2, X3) [1] a__g(X) → g(X) [1] a__b → b [1] The TRS has the following type information: a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f g :: g:b:c:f → g:b:c:f a__g :: g:b:c:f → g:b:c:f b :: g:b:c:f c :: g:b:c:f a__b :: g:b:c:f mark :: g:b:c:f → g:b:c:f f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f 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:

b => 0
c => 1

(10) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(Y, Y, Y) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = X
a__f(z, z', z'') -{ 1 }→ 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
mark(z) -{ 2 }→ a__g(a__g(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 2 }→ a__g(a__b) :|: z = 1 + 0
mark(z) -{ 2 }→ a__g(1) :|: z = 1 + 1
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ a__b :|: z = 0
mark(z) -{ 1 }→ 1 :|: z = 1

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

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

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(Y, Y, Y) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = X
a__f(z, z', z'') -{ 1 }→ 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
mark(z) -{ 2 }→ a__g(a__g(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 3 }→ a__g(1) :|: z = 1 + 0
mark(z) -{ 3 }→ a__g(0) :|: z = 1 + 0
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 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:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 3 }→ a__g(1) :|: z = 1 + 0
mark(z) -{ 3 }→ a__g(0) :|: z = 1 + 0
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

Found the following analysis order by SCC decomposition:

{ a__b } { a__g } { a__f } { mark }

(16) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 3 }→ a__g(1) :|: z = 1 + 0
mark(z) -{ 3 }→ a__g(0) :|: z = 1 + 0
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 3 }→ a__g(1) :|: z = 1 + 0
mark(z) -{ 3 }→ a__g(0) :|: z = 1 + 0
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

a__b: runtime: ?, size: O(1) [1]

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

Computed RUNTIME bound using CoFloCo for: a__b
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__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 3 }→ a__g(1) :|: z = 1 + 0
mark(z) -{ 3 }→ a__g(0) :|: z = 1 + 0
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

a__b: runtime: O(1) [1], size: O(1) [1]

(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__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 3 }→ a__g(1) :|: z = 1 + 0
mark(z) -{ 3 }→ a__g(0) :|: z = 1 + 0
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

a__b: runtime: O(1) [1], size: O(1) [1]

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 3 }→ a__g(1) :|: z = 1 + 0
mark(z) -{ 3 }→ a__g(0) :|: z = 1 + 0
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

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

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 3 }→ a__g(1) :|: z = 1 + 0
mark(z) -{ 3 }→ a__g(0) :|: z = 1 + 0
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

a__b: runtime: O(1) [1], size: O(1) [1] a__g: runtime: O(1) [1], size: O(n1) [1 + 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__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 1 + 1, z = 1 + 0
mark(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1, z = 1 + 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

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

(29) 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: 1 + z + z' + 3·z''

(30) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 1 + 1, z = 1 + 0
mark(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1, z = 1 + 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

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

(31) 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

(32) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 1 }→ a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 1 + 1, z = 1 + 0
mark(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1, z = 1 + 0
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 1 }→ a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

a__b: runtime: O(1) [1], size: O(1) [1] a__g: runtime: O(1) [1], size: O(n1) [1 + z] a__f: runtime: O(1) [2], size: O(n1) [1 + z + z' + 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__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 3 }→ s'' :|: s'' >= 0, s'' <= 1 * z'' + 1 * z'' + 3 * z'' + 1, z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 1 + 1, z = 1 + 0
mark(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1, z = 1 + 0
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * X1 + 1 * X2 + 3 * X3 + 1, X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 5 }→ s3 :|: s2 >= 0, s2 <= 1 * X1' + 1 * X2' + 3 * X3' + 1, s3 >= 0, s3 <= 1 * s2 + 1, X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

a__b: runtime: O(1) [1], size: O(1) [1] a__g: runtime: O(1) [1], size: O(n1) [1 + z] a__f: runtime: O(1) [2], size: O(n1) [1 + z + z' + 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: 1 + 3·z

(36) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 3 }→ s'' :|: s'' >= 0, s'' <= 1 * z'' + 1 * z'' + 3 * z'' + 1, z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 1 + 1, z = 1 + 0
mark(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1, z = 1 + 0
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * X1 + 1 * X2 + 3 * X3 + 1, X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 5 }→ s3 :|: s2 >= 0, s2 <= 1 * X1' + 1 * X2' + 3 * X3' + 1, s3 >= 0, s3 <= 1 * s2 + 1, X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

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

a__b: runtime: O(1) [1], size: O(1) [1] a__g: runtime: O(1) [1], size: O(n1) [1 + z] a__f: runtime: O(1) [2], size: O(n1) [1 + z + z' + 3·z''] mark: runtime: ?, size: O(n1) [1 + 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: 22 + 4·z

(38) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z', z'') -{ 3 }→ s'' :|: s'' >= 0, s'' <= 1 * z'' + 1 * z'' + 3 * z'' + 1, z'' >= 0, z' - 1 >= 0, z = z' - 1
a__f(z, z', z'') -{ 1 }→ 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
a__g(z) -{ 1 }→ 1 :|: z = 0
a__g(z) -{ 1 }→ 1 + z :|: z >= 0
mark(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 1 + 1, z = 1 + 0
mark(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1, z = 1 + 0
mark(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * X1 + 1 * X2 + 3 * X3 + 1, X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
mark(z) -{ 5 }→ s3 :|: s2 >= 0, s2 <= 1 * X1' + 1 * X2' + 3 * X3' + 1, s3 >= 0, s3 <= 1 * s2 + 1, X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
mark(z) -{ 2 }→ a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
mark(z) -{ 1 }→ 1 :|: z = 1
mark(z) -{ 2 }→ 1 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 0
mark(z) -{ 3 }→ 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed:
Previous analysis results are:

a__b: runtime: O(1) [1], size: O(1) [1] a__g: runtime: O(1) [1], size: O(n1) [1 + z] a__f: runtime: O(1) [2], size: O(n1) [1 + z + z' + 3·z''] mark: runtime: O(n1) [22 + 4·z], size: O(n1) [1 + 3·z]

(39) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(40) BOUNDS(1, n^1)