YES Termination w.r.t. Q proof of Waldmann_06_jwteparla1.ari

(0) Obligation:

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

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

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

The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))

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

(5) TransformationProof (EQUIVALENT transformation)

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

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

F(x, f(a, f(f(a, a), a))) → F(a, x)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))

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

(9) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(10) Obligation:

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

F(x, f(a, f(f(a, a), a))) → F(a, x)

R is empty.
The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))

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

(11) TransformationProof (EQUIVALENT transformation)

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

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

(12) Obligation:

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

F(a, f(a, f(f(a, a), a))) → F(a, a)

R is empty.
The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))

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