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(h(x)) → f(i(x))
g(i(x)) → g(h(x))
h(a) → b
i(a) → b

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(h(z0)) → f(i(z0))
g(i(z0)) → g(h(z0))
h(a) → b
i(a) → b

Tuples:

F(h(z0)) → c(F(i(z0)), I(z0))
G(i(z0)) → c1(G(h(z0)), H(z0))

S tuples:

F(h(z0)) → c(F(i(z0)), I(z0))
G(i(z0)) → c1(G(h(z0)), H(z0))

K tuples:none
Defined Rule Symbols:

f, g, h, i

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(h(z0)) → c(F(i(z0)), I(z0))
G(i(z0)) → c1(G(h(z0)), H(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(h(z0)) → f(i(z0))
g(i(z0)) → g(h(z0))
h(a) → b
i(a) → b

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

f, g, h, i

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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