### (0) Obligation:

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(f(a)) → a__f(g(f(a)))
a__f(z0) → f(z0)
mark(f(z0)) → a__f(z0)
mark(a) → a
mark(g(z0)) → g(mark(z0))
Tuples:

A__F(f(a)) → c(A__F(g(f(a))))
A__F(z0) → c1
MARK(f(z0)) → c2(A__F(z0))
MARK(a) → c3
MARK(g(z0)) → c4(MARK(z0))
S tuples:

A__F(f(a)) → c(A__F(g(f(a))))
A__F(z0) → c1
MARK(f(z0)) → c2(A__F(z0))
MARK(a) → c3
MARK(g(z0)) → c4(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

A__F, MARK

Compound Symbols:

c, c1, c2, c3, c4

### (3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

A__F(f(a)) → c(A__F(g(f(a))))
A__F(z0) → c1
MARK(f(z0)) → c2(A__F(z0))
MARK(a) → c3

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(f(a)) → a__f(g(f(a)))
a__f(z0) → f(z0)
mark(f(z0)) → a__f(z0)
mark(a) → a
mark(g(z0)) → g(mark(z0))
Tuples:

MARK(g(z0)) → c4(MARK(z0))
S tuples:

MARK(g(z0)) → c4(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c4

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

MARK(g(z0)) → c4(MARK(z0))
S tuples:

MARK(g(z0)) → c4(MARK(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

MARK

Compound Symbols:

c4

### (7) CdtRuleRemovalProof (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(g(z0)) → c4(MARK(z0))
We considered the (Usable) Rules:none
And the Tuples:

MARK(g(z0)) → c4(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(MARK(x1)) = [5]x1
POL(c4(x1)) = x1
POL(g(x1)) = [1] + x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

MARK(g(z0)) → c4(MARK(z0))
S tuples:none
K tuples:

MARK(g(z0)) → c4(MARK(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

MARK

Compound Symbols:

c4

### (9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty