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

0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 UsableRulesProof (⇔)
↳8 QDP
↳9 QReductionProof (⇔)
↳10 QDP
↳11 NonTerminationProof (⇔)
↳12 FALSE

(0) Obligation:

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

f(g(X), Y) → f(X, f(g(X), Y))

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(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

f(g(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(g(X), Y) → F(X, f(g(X), Y))
F(g(X), Y) → F(g(X), Y)

The TRS R consists of the following rules:

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

f(g(x0), x1)

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F(g(X), Y) → F(X, f(g(X), Y))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(F(x1, x2)) =
/1\
\0/
 +
/01\
\00/
·x1 +
/00\
\00/
·x2

POL(g(x1)) =
/0\
\1/
 +
/00\
\01/
·x1

POL(f(x1, x2)) =
/0\
\0/
 +
/11\
\10/
·x1 +
/01\
\10/
·x2

The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

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

F(g(X), Y) → F(g(X), Y)

The TRS R consists of the following rules:

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

f(g(x0), x1)

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(g(X), Y) → F(g(X), Y)

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

f(g(x0), x1)

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].

f(g(x0), x1)

(10) Obligation:

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

F(g(X), Y) → F(g(X), Y)

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 = F(g(X), Y) evaluates to t =F(g(X), Y)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from F(g(X), Y) to F(g(X), Y).



(12) FALSE