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(x1) → x1
a(a(b(x1))) → b(a(b(a(c(a(x1))))))
c(b(x1)) → x1

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(a(b(z0))) → b(a(b(a(c(a(z0))))))
c(b(z0)) → z0

Tuples:

A(a(b(z0))) → c2(A(b(a(c(a(z0))))), A(c(a(z0))), C(a(z0)), A(z0))

S tuples:

A(a(b(z0))) → c2(A(b(a(c(a(z0))))), A(c(a(z0))), C(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A

Compound Symbols:

c2

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

A(a(b(z0))) → c2(A(b(a(c(a(z0))))), A(c(a(z0))), C(a(z0)), A(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(a(b(z0))) → b(a(b(a(c(a(z0))))))
c(b(z0)) → z0

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

a, c

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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