The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(f(x)) → g(f(x))
g(g(x)) → f(x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → g(f(z0))
g(g(z0)) → f(z0)

Tuples:

F(f(z0)) → c(G(f(z0)), F(z0))
G(g(z0)) → c1(F(z0))

S tuples:

F(f(z0)) → c(G(f(z0)), F(z0))
G(g(z0)) → c1(F(z0))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

G(g(z0)) → c1(F(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → g(f(z0))
g(g(z0)) → f(z0)

Tuples:

F(f(z0)) → c(G(f(z0)), F(z0))

S tuples:

F(f(z0)) → c(G(f(z0)), F(z0))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(f(z0)) → c(G(f(z0)), F(z0))

We considered the (Usable) Rules:none
And the Tuples:

F(f(z0)) → c(G(f(z0)), F(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x₁)) = [2]x₁²   
POL(G(x₁)) = 0   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(f(x₁)) = [2] + [3]x₁²   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → g(f(z0))
g(g(z0)) → f(z0)

Tuples:

F(f(z0)) → c(G(f(z0)), F(z0))

S tuples:none
K tuples:

F(f(z0)) → c(G(f(z0)), F(z0))

Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))