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:

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

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

b, a

Defined Pair Symbols:

B, A

Compound Symbols:

c1, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

b, a

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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