### (0) Obligation:

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

a__f(b, X, c) → a__f(X, a__c, X)
a__cb
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(c) → a__c
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__cc

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(b, z0, c) → a__f(z0, a__c, z0)
a__f(z0, z1, z2) → f(z0, z1, z2)
a__cb
a__cc
mark(f(z0, z1, z2)) → a__f(z0, mark(z1), z2)
mark(c) → a__c
mark(b) → b
Tuples:

A__F(b, z0, c) → c1(A__F(z0, a__c, z0), A__C)
A__F(z0, z1, z2) → c2
A__Cc3
A__Cc4
MARK(f(z0, z1, z2)) → c5(A__F(z0, mark(z1), z2), MARK(z1))
MARK(c) → c6(A__C)
MARK(b) → c7
S tuples:

A__F(b, z0, c) → c1(A__F(z0, a__c, z0), A__C)
A__F(z0, z1, z2) → c2
A__Cc3
A__Cc4
MARK(f(z0, z1, z2)) → c5(A__F(z0, mark(z1), z2), MARK(z1))
MARK(c) → c6(A__C)
MARK(b) → c7
K tuples:none
Defined Rule Symbols:

a__f, a__c, mark

Defined Pair Symbols:

A__F, A__C, MARK

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7

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

Removed 6 trailing nodes:

A__F(b, z0, c) → c1(A__F(z0, a__c, z0), A__C)
A__F(z0, z1, z2) → c2
A__Cc3
A__Cc4
MARK(c) → c6(A__C)
MARK(b) → c7

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(b, z0, c) → a__f(z0, a__c, z0)
a__f(z0, z1, z2) → f(z0, z1, z2)
a__cb
a__cc
mark(f(z0, z1, z2)) → a__f(z0, mark(z1), z2)
mark(c) → a__c
mark(b) → b
Tuples:

MARK(f(z0, z1, z2)) → c5(A__F(z0, mark(z1), z2), MARK(z1))
S tuples:

MARK(f(z0, z1, z2)) → c5(A__F(z0, mark(z1), z2), MARK(z1))
K tuples:none
Defined Rule Symbols:

a__f, a__c, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c5

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

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(b, z0, c) → a__f(z0, a__c, z0)
a__f(z0, z1, z2) → f(z0, z1, z2)
a__cb
a__cc
mark(f(z0, z1, z2)) → a__f(z0, mark(z1), z2)
mark(c) → a__c
mark(b) → b
Tuples:

MARK(f(z0, z1, z2)) → c5(MARK(z1))
S tuples:

MARK(f(z0, z1, z2)) → c5(MARK(z1))
K tuples:none
Defined Rule Symbols:

a__f, a__c, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c5

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

a__f(b, z0, c) → a__f(z0, a__c, z0)
a__f(z0, z1, z2) → f(z0, z1, z2)
a__cb
a__cc
mark(f(z0, z1, z2)) → a__f(z0, mark(z1), z2)
mark(c) → a__c
mark(b) → b

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

MARK(f(z0, z1, z2)) → c5(MARK(z1))
S tuples:

MARK(f(z0, z1, z2)) → c5(MARK(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

MARK

Compound Symbols:

c5

### (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(f(z0, z1, z2)) → c5(MARK(z1))
We considered the (Usable) Rules:none
And the Tuples:

MARK(f(z0, z1, z2)) → c5(MARK(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(MARK(x1)) = [5]x1
POL(c5(x1)) = x1
POL(f(x1, x2, x3)) = [1] + x2

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

MARK(f(z0, z1, z2)) → c5(MARK(z1))
S tuples:none
K tuples:

MARK(f(z0, z1, z2)) → c5(MARK(z1))
Defined Rule Symbols:none

Defined Pair Symbols:

MARK

Compound Symbols:

c5

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

The set S is empty