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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))