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(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(b, z0)) → f(b, f(a, z0))
f(b, f(c, z0)) → f(c, f(b, z0))
f(c, f(a, z0)) → f(a, f(c, z0))

Tuples:

F(a, f(b, z0)) → c1(F(b, f(a, z0)), F(a, z0))
F(b, f(c, z0)) → c2(F(c, f(b, z0)), F(b, z0))
F(c, f(a, z0)) → c3(F(a, f(c, z0)), F(c, z0))

S tuples:

F(a, f(b, z0)) → c1(F(b, f(a, z0)), F(a, z0))
F(b, f(c, z0)) → c2(F(c, f(b, z0)), F(b, z0))
F(c, f(a, z0)) → c3(F(a, f(c, z0)), F(c, z0))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

F(a, f(b, z0)) → c1(F(b, f(a, z0)), F(a, z0))
F(b, f(c, z0)) → c2(F(c, f(b, z0)), F(b, z0))
F(c, f(a, z0)) → c3(F(a, f(c, z0)), F(c, z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(b, z0)) → f(b, f(a, z0))
f(b, f(c, z0)) → f(c, f(b, z0))
f(c, f(a, z0)) → f(a, f(c, z0))

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

f

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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