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:

eq → true
eq → eq
eq → false
inf(X) → cons
take(0, X) → nil
take(s, cons) → cons
length(nil) → 0
length(cons) → s

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Knuth-Bendix order [KBO] with precedence:

eq > true
eq > false

and weight map:

eq=3
inf_1=3
cons=5
true=3
false=3
s=6
take_2=3
length_1=4
0=8
nil=8

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

eq → true
eq → false
inf(X) → cons
take(0, X) → nil
take(s, cons) → cons
length(nil) → 0
length(cons) → s

(2) Obligation:

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

eq → eq

Q is empty.

(3) AAECC Innermost (EQUIVALENT transformation)

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

The TRS R 2 is

eq → eq

The signature Sigma is {eq}

(4) Obligation:

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

eq → eq

The set Q consists of the following terms:

eq

(5) DependencyPairsProof (EQUIVALENT transformation)

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

(6) Obligation:

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

EQ → EQ

The TRS R consists of the following rules:

eq → eq

The set Q consists of the following terms:

eq

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:

EQ → EQ

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

eq

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

(9) 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].

eq

(10) Obligation:

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

EQ → EQ

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

(11) 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 = EQ evaluates to t =EQ

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from EQ to EQ.

(12) NO