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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(a, h(z0)) → c(F(g(z0), h(z0)), G(z0), H(z0))
H(g(z0)) → c1(H(a))
G(h(z0)) → c3(G(z0))

S tuples:

F(a, h(z0)) → c(F(g(z0), h(z0)), G(z0), H(z0))
H(g(z0)) → c1(H(a))
G(h(z0)) → c3(G(z0))

K tuples:none
Defined Rule Symbols:

f, h, g

Defined Pair Symbols:

F, H, G

Compound Symbols:

c, c1, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(a, h(z0)) → c(F(g(z0), h(z0)), G(z0), H(z0))
H(g(z0)) → c1(H(a))
G(h(z0)) → c3(G(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

f, h, g

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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