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(g(x), x, y) → f(y, y, g(y))
g(g(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(g(z0), z0, z1) → f(z1, z1, g(z1))
g(g(z0)) → g(z0)

Tuples:

F(g(z0), z0, z1) → c(F(z1, z1, g(z1)), G(z1))
G(g(z0)) → c1(G(z0))

S tuples:

F(g(z0), z0, z1) → c(F(z1, z1, g(z1)), G(z1))
G(g(z0)) → c1(G(z0))

K tuples:none
Defined Rule Symbols:

f, g

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(g(z0), z0, z1) → c(F(z1, z1, g(z1)), G(z1))
G(g(z0)) → c1(G(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(g(z0), z0, z1) → f(z1, z1, g(z1))
g(g(z0)) → g(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))