(0) Obligation:

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

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

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

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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