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

(0) Obligation:

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

f(s(x)) → f(g(x, x))
g(0, 1) → s(0)
0 → 1

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(s(z0)) → f(g(z0, z0))
g(0, 1) → s(0)
0 → 1

Tuples:

F(s(z0)) → c(F(g(z0, z0)), G(z0, z0))
G(0, 1) → c1(0')
0' → c2

S tuples:

F(s(z0)) → c(F(g(z0, z0)), G(z0, z0))
G(0, 1) → c1(0')
0' → c2

K tuples:none
Defined Rule Symbols:

f, g, 0

Defined Pair Symbols:

F, G, 0'

Compound Symbols:

c, c1, c2

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

Removed 3 trailing nodes:

F(s(z0)) → c(F(g(z0, z0)), G(z0, z0))
0' → c2
G(0, 1) → c1(0')

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(s(z0)) → f(g(z0, z0))
g(0, 1) → s(0)
0 → 1

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

f, g, 0

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(6) BOUNDS(1, 1)