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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

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:

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))