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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

C(c(a(z0))) → c3(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))

S tuples:

C(c(a(z0))) → c3(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C

Compound Symbols:

c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

C(c(a(z0))) → c3(A(c(a(c(c(z0))))), C(a(c(c(z0)))), A(c(c(z0))), C(c(z0)), C(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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