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(b(x)) → b(a(x))
a(c(x)) → x

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS 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))

S tuples:

A(b(z0)) → c1(A(z0))

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1

(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.

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   

(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:none
K tuples:

A(b(z0)) → c1(A(z0))

Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))