### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

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

A__F(f(a)) → c1
A__F(z0) → c2
MARK(f(z0)) → c3(A__F(mark(z0)), MARK(z0))
MARK(a) → c4
MARK(c(z0)) → c5
MARK(g(z0)) → c6(MARK(z0))
S tuples:

A__F(f(a)) → c1
A__F(z0) → c2
MARK(f(z0)) → c3(A__F(mark(z0)), MARK(z0))
MARK(a) → c4
MARK(c(z0)) → c5
MARK(g(z0)) → c6(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

A__F, MARK

Compound Symbols:

c1, c2, c3, c4, c5, c6

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

Removed 4 trailing nodes:

A__F(f(a)) → c1
A__F(z0) → c2
MARK(a) → c4
MARK(c(z0)) → c5

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

MARK(f(z0)) → c3(A__F(mark(z0)), MARK(z0))
MARK(g(z0)) → c6(MARK(z0))
S tuples:

MARK(f(z0)) → c3(A__F(mark(z0)), MARK(z0))
MARK(g(z0)) → c6(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c3, c6

### (5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

MARK(g(z0)) → c6(MARK(z0))
MARK(f(z0)) → c3(MARK(z0))
S tuples:

MARK(g(z0)) → c6(MARK(z0))
MARK(f(z0)) → c3(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c6, c3

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

MARK(g(z0)) → c6(MARK(z0))
MARK(f(z0)) → c3(MARK(z0))
S tuples:

MARK(g(z0)) → c6(MARK(z0))
MARK(f(z0)) → c3(MARK(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

MARK

Compound Symbols:

c6, c3

### (9) 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)) → c6(MARK(z0))
MARK(f(z0)) → c3(MARK(z0))
We considered the (Usable) Rules:none
And the Tuples:

MARK(g(z0)) → c6(MARK(z0))
MARK(f(z0)) → c3(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(MARK(x1)) = [4]x1
POL(c3(x1)) = x1
POL(c6(x1)) = x1
POL(f(x1)) = [4] + x1
POL(g(x1)) = [1] + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

MARK(g(z0)) → c6(MARK(z0))
MARK(f(z0)) → c3(MARK(z0))
S tuples:none
K tuples:

MARK(g(z0)) → c6(MARK(z0))
MARK(f(z0)) → c3(MARK(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

MARK

Compound Symbols:

c6, c3

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

The set S is empty