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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A(a(z0)) → c1(B(z0))
A(b(z0)) → c2(A(z0))
B(c(z0)) → c3(B(a(z0)), A(z0))

S tuples:

A(a(z0)) → c1(B(z0))
A(b(z0)) → c2(A(z0))
B(c(z0)) → c3(B(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

A, B

Compound Symbols:

c1, c2, c3

(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace B(c(z0)) → c3(B(a(z0)), A(z0)) by

B(c(a(z0))) → c3(B(b(z0)), A(a(z0)))
B(c(b(z0))) → c3(B(c(a(z0))), A(b(z0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A(a(z0)) → c1(B(z0))
A(b(z0)) → c2(A(z0))
B(c(a(z0))) → c3(B(b(z0)), A(a(z0)))
B(c(b(z0))) → c3(B(c(a(z0))), A(b(z0)))

S tuples:

A(a(z0)) → c1(B(z0))
A(b(z0)) → c2(A(z0))
B(c(a(z0))) → c3(B(b(z0)), A(a(z0)))
B(c(b(z0))) → c3(B(c(a(z0))), A(b(z0)))

K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

A, B

Compound Symbols:

c1, c2, c3

(5) CdtUnreachableProof (EQUIVALENT transformation)

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

A(a(z0)) → c1(B(z0))
A(b(z0)) → c2(A(z0))
B(c(a(z0))) → c3(B(b(z0)), A(a(z0)))
B(c(b(z0))) → c3(B(c(a(z0))), A(b(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

a, b

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))