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:

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

G(h(g(z0))) → c(G(z0))
G(g(z0)) → c1(G(h(g(z0))), H(g(z0)), G(z0))
H(h(z0)) → c2(H(f(h(z0), z0)), H(z0))

S tuples:

G(h(g(z0))) → c(G(z0))
G(g(z0)) → c1(G(h(g(z0))), H(g(z0)), G(z0))
H(h(z0)) → c2(H(f(h(z0), z0)), H(z0))

K tuples:none
Defined Rule Symbols:

g, h

Defined Pair Symbols:

G, H

Compound Symbols:

c, c1, c2

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

G(h(g(z0))) → c(G(z0))
G(g(z0)) → c1(G(h(g(z0))), H(g(z0)), G(z0))
H(h(z0)) → c2(H(f(h(z0), z0)), H(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

g, h

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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