The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Runtime Complexity TRS:
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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(z0) → g(h(f(z0)))
a__f(z0) → f(z0)
mark(f(z0)) → a__f(mark(z0))
mark(g(z0)) → g(z0)
mark(h(z0)) → h(mark(z0))

Tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c4(MARK(z0))

S tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c4(MARK(z0))

K tuples:none
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c2, c4

(3) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(h(z0)) → c4(MARK(z0))

We considered the (Usable) Rules:

mark(f(z0)) → a__f(mark(z0))
mark(g(z0)) → g(z0)
mark(h(z0)) → h(mark(z0))
a__f(z0) → g(h(f(z0)))
a__f(z0) → f(z0)

And the Tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c4(MARK(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A__F(x₁)) = 0   
POL(MARK(x₁)) = [2]x₁   
POL(a__f(x₁)) = [2]x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁)) = x₁   
POL(f(x₁)) = x₁   
POL(g(x₁)) = [3]   
POL(h(x₁)) = [1] + x₁   
POL(mark(x₁)) = [3]   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(z0) → g(h(f(z0)))
a__f(z0) → f(z0)
mark(f(z0)) → a__f(mark(z0))
mark(g(z0)) → g(z0)
mark(h(z0)) → h(mark(z0))

Tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c4(MARK(z0))

S tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))

K tuples:

MARK(h(z0)) → c4(MARK(z0))

Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c2, c4

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))

We considered the (Usable) Rules:

mark(f(z0)) → a__f(mark(z0))
mark(g(z0)) → g(z0)
mark(h(z0)) → h(mark(z0))
a__f(z0) → g(h(f(z0)))
a__f(z0) → f(z0)

And the Tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c4(MARK(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A__F(x₁)) = 0   
POL(MARK(x₁)) = x₁   
POL(a__f(x₁)) = [4] + [2]x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁)) = x₁   
POL(f(x₁)) = [2] + x₁   
POL(g(x₁)) = [3]   
POL(h(x₁)) = [3] + x₁   
POL(mark(x₁)) = [4]x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(z0) → g(h(f(z0)))
a__f(z0) → f(z0)
mark(f(z0)) → a__f(mark(z0))
mark(g(z0)) → g(z0)
mark(h(z0)) → h(mark(z0))

Tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c4(MARK(z0))

S tuples:none
K tuples:

MARK(h(z0)) → c4(MARK(z0))
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))

Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c2, c4

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))