(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(a, f(a, x)) → f(x, f(a, f(f(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(a, f(a, z0)) → f(z0, f(a, f(f(a, a), a)))
Tuples:

F(a, f(a, z0)) → c(F(z0, f(a, f(f(a, a), a))), F(a, f(f(a, a), a)), F(f(a, a), a), F(a, a))
S tuples:

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, z0)) → f(z0, f(a, f(f(a, a), a)))
Tuples:

F(a, f(a, x0)) → c(F(x0, f(a, f(f(a, a), a))))
S tuples:

F(a, f(a, x0)) → c(F(x0, f(a, f(f(a, a), a))))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use instantiation to replace F(a, f(a, x0)) → c(F(x0, f(a, f(f(a, a), a)))) by

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, z0)) → f(z0, f(a, f(f(a, a), a)))
Tuples:

F(a, f(a, f(f(a, a), a))) → c(F(f(f(a, a), a), f(a, f(f(a, a), a))))
S tuples:

F(a, f(a, f(f(a, a), a))) → c(F(f(f(a, a), a), f(a, f(f(a, a), a))))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, z0)) → f(z0, f(a, f(f(a, a), a)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:none

Compound Symbols:none

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))