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:

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

H(z0, z1) → c(F(z0, z1, z0))
F(0, 1, z0) → c1(H(z0, z0))

S tuples:

H(z0, z1) → c(F(z0, z1, z0))
F(0, 1, z0) → c1(H(z0, z0))

K tuples:none
Defined Rule Symbols:

h, f, g

Defined Pair Symbols:

H, F

Compound Symbols:

c, c1

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

Use instantiation to replace H(z0, z1) → c(F(z0, z1, z0)) by

H(x0, x0) → c(F(x0, x0, x0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(0, 1, z0) → c1(H(z0, z0))
H(x0, x0) → c(F(x0, x0, x0))

S tuples:

F(0, 1, z0) → c1(H(z0, z0))
H(x0, x0) → c(F(x0, x0, x0))

K tuples:none
Defined Rule Symbols:

h, f, g

Defined Pair Symbols:

F, H

Compound Symbols:

c1, c

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

Removed 2 trailing nodes:

F(0, 1, z0) → c1(H(z0, z0))
H(x0, x0) → c(F(x0, x0, x0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

h, f, g

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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