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(x, y, z) → g(x, y, z)
g(0, 1, x) → f(x, x, x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z1, z2) → g(z0, z1, z2)
g(0, 1, z0) → f(z0, z0, z0)

Tuples:

F(z0, z1, z2) → c(G(z0, z1, z2))
G(0, 1, z0) → c1(F(z0, z0, z0))

S tuples:

F(z0, z1, z2) → c(G(z0, z1, z2))
G(0, 1, z0) → c1(F(z0, z0, z0))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1

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

Use forward instantiation to replace F(z0, z1, z2) → c(G(z0, z1, z2)) by

F(0, 1, z2) → c(G(0, 1, z2))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z1, z2) → g(z0, z1, z2)
g(0, 1, z0) → f(z0, z0, z0)

Tuples:

G(0, 1, z0) → c1(F(z0, z0, z0))
F(0, 1, z2) → c(G(0, 1, z2))

S tuples:

G(0, 1, z0) → c1(F(z0, z0, z0))
F(0, 1, z2) → c(G(0, 1, z2))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c

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

Removed 2 trailing nodes:

F(0, 1, z2) → c(G(0, 1, z2))
G(0, 1, z0) → c1(F(z0, z0, z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z1, z2) → g(z0, z1, z2)
g(0, 1, z0) → f(z0, z0, z0)

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))