(0) Obligation:

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

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

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

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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