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(f(x)) → f(x)
g(0) → g(f(0))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(z0)
g(0) → g(f(0))

Tuples:

F(f(z0)) → c(F(z0))
G(0) → c1(G(f(0)), F(0))

S tuples:

F(f(z0)) → c(F(z0))
G(0) → c1(G(f(0)), F(0))

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(f(z0)) → c(F(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(z0)
g(0) → g(f(0))

Tuples:

G(0) → c1(G(f(0)), F(0))

S tuples:

G(0) → c1(G(f(0)), F(0))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

G

Compound Symbols:

c1

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

Removed 1 trailing nodes:

G(0) → c1(G(f(0)), F(0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(z0)
g(0) → g(f(0))

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

f, g

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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