YES Termination w.r.t. Q proof of Zantema_05_jw30.ari

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(a, a), x0)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(f(a, a), x) → F(f(x, a), f(a, f(a, a)))
F(f(a, a), x) → F(x, a)
F(f(a, a), x) → F(a, f(a, a))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(a, a), x0)

We have to consider all minimal (P,Q,R)-chains.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(f(a, a), x) → F(x, a)
F(f(a, a), x) → F(f(x, a), f(a, f(a, a)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(a, a), x0)

We have to consider all minimal (P,Q,R)-chains.

(7) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule F(f(a, a), x) → F(x, a) we obtained the following new rules [LPAR04]:

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

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(f(a, a), x) → F(f(x, a), f(a, f(a, a)))
F(f(a, a), a) → F(a, a)
F(f(a, a), f(a, f(a, a))) → F(f(a, f(a, a)), a)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(a, a), x0)

We have to consider all minimal (P,Q,R)-chains.

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(f(a, a), x) → F(f(x, a), f(a, f(a, a)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(a, a), x0)

We have to consider all minimal (P,Q,R)-chains.

(11) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule F(f(a, a), x) → F(f(x, a), f(a, f(a, a))) we obtained the following new rules [LPAR04]:

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

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(a, a), x0)

We have to consider all minimal (P,Q,R)-chains.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(14) TRUE