### (0) Obligation:

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

a(b(x)) → b(a(x))
a(c(x)) → 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(b(z0)) → b(a(z0))
a(c(z0)) → z0
Tuples:

A(b(z0)) → c1(A(z0))
A(c(z0)) → c2
S tuples:

A(b(z0)) → c1(A(z0))
A(c(z0)) → c2
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1, c2

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

Removed 1 trailing nodes:

A(c(z0)) → c2

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(z0))
a(c(z0)) → z0
Tuples:

A(b(z0)) → c1(A(z0))
S tuples:

A(b(z0)) → c1(A(z0))
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

a(b(z0)) → b(a(z0))
a(c(z0)) → z0

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

A(b(z0)) → c1(A(z0))
S tuples:

A(b(z0)) → c1(A(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

A

Compound Symbols:

c1

### (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.

A(b(z0)) → c1(A(z0))
We considered the (Usable) Rules:none
And the Tuples:

A(b(z0)) → c1(A(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A(x1)) = [5]x1
POL(b(x1)) = [1] + x1
POL(c1(x1)) = x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

A(b(z0)) → c1(A(z0))
S tuples:none
K tuples:

A(b(z0)) → c1(A(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

A

Compound Symbols:

c1

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

The set S is empty