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, f(a, a)), a), x) → f(x, f(x, a))

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, f(a, a)), a), z0) → f(z0, f(z0, a))

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(f(f(a, f(a, a)), a), z0) → c(F(z0, f(z0, a)), F(z0, a)) by

F(f(f(a, f(a, a)), a), f(f(a, f(a, a)), a)) → c(F(f(f(a, f(a, a)), a), f(a, f(a, a))), F(f(f(a, f(a, a)), a), a))
F(f(f(a, f(a, a)), a), x0) → c

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(f(f(a, f(a, a)), a), f(f(a, f(a, a)), a)) → c(F(f(f(a, f(a, a)), a), f(a, f(a, a))), F(f(f(a, f(a, a)), a), a))
F(f(f(a, f(a, a)), a), x0) → c

S tuples:

F(f(f(a, f(a, a)), a), f(f(a, f(a, a)), a)) → c(F(f(f(a, f(a, a)), a), f(a, f(a, a))), F(f(f(a, f(a, a)), a), a))
F(f(f(a, f(a, a)), a), x0) → c

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Removed 2 trailing nodes:

F(f(f(a, f(a, a)), a), x0) → c
F(f(f(a, f(a, a)), a), f(f(a, f(a, a)), a)) → c(F(f(f(a, f(a, a)), a), f(a, f(a, a))), F(f(f(a, f(a, a)), a), a))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

f

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))