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:

c(b(a(X))) → a(a(b(b(c(c(X))))))
a(X) → e
b(X) → e
c(X) → e

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

c, a, b

Defined Pair Symbols:

C

Compound Symbols:

c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

c, a, b

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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