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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))