YES Termination w.r.t. Q proof of Zantema_05_jw42.ari

(0) Obligation:

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

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

The set Q consists of the following terms:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(a, x0), a)

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 3 less nodes.

(6) Obligation:

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

F(f(a, x), a) → F(f(a, f(a, a)), x)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(a, x0), a)

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

(7) 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.

(8) Obligation:

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

F(f(a, x), a) → F(f(a, f(a, a)), x)

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

f(f(a, x0), a)

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


F(f(a, x), a) → F(f(a, f(a, a)), x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO,RATPOLO]:

POL(F(x1, x2)) = x1 + [1/4]x2   
POL(a) = [1/2]   
POL(f(x1, x2)) = [1/4]x2   
The value of delta used in the strict ordering is 3/32.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(10) Obligation:

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

f(f(a, x0), a)

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

(11) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(12) YES