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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c, c1

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

Removed 2 trailing nodes:

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z1) → h(z0, z1)
f(z0, z1) → h(z1, z0)
h(z0, z0) → z0

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

f, h

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))