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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(f(a, z0), z1), z2) → f(f(z0, z2), f(z1, z2))
f(f(b, z0), z1) → z0
f(c, z0) → z0

Tuples:

F(f(f(a, z0), z1), z2) → c1(F(f(z0, z2), f(z1, z2)), F(z0, z2), F(z1, z2))

S tuples:

F(f(f(a, z0), z1), z2) → c1(F(f(z0, z2), f(z1, z2)), F(z0, z2), F(z1, z2))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(f(a, z0), z1), z2) → c1(F(f(z0, z2), f(z1, z2)), F(z0, z2), F(z1, z2))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(f(a, z0), z1), z2) → f(f(z0, z2), f(z1, z2))
f(f(b, z0), z1) → z0
f(c, z0) → 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))