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) → f(b)
g(b) → g(a)
f(x) → g(x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → f(b)
f(z0) → g(z0)
g(b) → g(a)

Tuples:

F(a) → c(F(b))
F(z0) → c1(G(z0))
G(b) → c2(G(a))

S tuples:

F(a) → c(F(b))
F(z0) → c1(G(z0))
G(b) → c2(G(a))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

F(a) → c(F(b))
G(b) → c2(G(a))
F(z0) → c1(G(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → f(b)
f(z0) → g(z0)
g(b) → g(a)

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

f, g

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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