Termination w.r.t. Q of the given QTRS could be disproven:

(0) Obligation:

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

f(x, y) → f(x, x)
f(s(x), y) → f(y, x)

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none

The TRS R 2 is

f(x, y) → f(x, x)
f(s(x), y) → f(y, x)

The signature Sigma is {f}

(2) Obligation:

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

f(x, y) → f(x, x)
f(s(x), y) → f(y, x)

The set Q consists of the following terms:

f(x0, x1)

(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, y) → F(x, x)
F(s(x), y) → F(y, x)

The TRS R consists of the following rules:

f(x, y) → f(x, x)
f(s(x), y) → f(y, x)

The set Q consists of the following terms:

f(x0, x1)

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

(5) 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.

(6) Obligation:

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

F(x, y) → F(x, x)
F(s(x), y) → F(y, x)

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

f(x0, x1)

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

(7) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

f(x0, x1)

(8) Obligation:

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

F(x, y) → F(x, x)
F(s(x), y) → F(y, x)

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

(9) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = F(x, y) evaluates to t =F(x, x)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [y / x]
• Semiunifier: [ ]

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Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from F(x, y) to F(x, x).

(10) NO