The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

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

a(a(f(b, a(x)))) → f(a(a(a(x))), b)
a(a(x)) → f(b, a(f(a(x), b)))
f(a(x), b) → f(b, a(x))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(f(b, a(z0)))) → f(a(a(a(z0))), b)
a(a(z0)) → f(b, a(f(a(z0), b)))
f(a(z0), b) → f(b, a(z0))

Tuples:

A(a(f(b, a(z0)))) → c(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0))
A(a(z0)) → c1(F(b, a(f(a(z0), b))), A(f(a(z0), b)), F(a(z0), b), A(z0))
F(a(z0), b) → c2(F(b, a(z0)), A(z0))

S tuples:

A(a(f(b, a(z0)))) → c(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0))
A(a(z0)) → c1(F(b, a(f(a(z0), b))), A(f(a(z0), b)), F(a(z0), b), A(z0))
F(a(z0), b) → c2(F(b, a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, f

Defined Pair Symbols:

A, F

Compound Symbols:

c, c1, c2

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

A(a(f(b, a(z0)))) → c(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0))
A(a(z0)) → c1(F(b, a(f(a(z0), b))), A(f(a(z0), b)), F(a(z0), b), A(z0))
F(a(z0), b) → c2(F(b, a(z0)), A(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(f(b, a(z0)))) → f(a(a(a(z0))), b)
a(a(z0)) → f(b, a(f(a(z0), b)))
f(a(z0), b) → f(b, a(z0))

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

a, f

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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