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:

f(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(g(f(g(f(z0)))))
f(g(f(z0))) → f(g(z0))

Tuples:

F(f(z0)) → c(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
F(g(f(z0))) → c1(F(g(z0)))

S tuples:

F(f(z0)) → c(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
F(g(f(z0))) → c1(F(g(z0)))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(z0)) → c(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
F(g(f(z0))) → c1(F(g(z0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(g(f(g(f(z0)))))
f(g(f(z0))) → f(g(z0))

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

f

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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