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:

h(f(f(x))) → h(f(g(f(x))))
f(g(f(x))) → f(f(x))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

h, f

Defined Pair Symbols:

H, F

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

h, f

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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