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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 QDPSizeChangeProof (⇔)
↳8 TRUE

(0) Obligation:

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

f(x, f(a, y)) → f(a, f(f(f(a, x), h(a)), y))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

The TRS R consists of the following rules:

f(x, f(a, y)) → f(a, f(f(f(a, x), h(a)), y))

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Obligation:

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

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

The TRS R consists of the following rules:

f(x, f(a, y)) → f(a, f(f(f(a, x), h(a)), y))

Q is empty.
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(x, f(a, y)) → F(a, f(f(f(a, x), h(a)), y))
F(x, f(a, y)) → F(a, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(F(x1, x2)) =
/1\
\0/
 +
/01\
\11/
·x1 +
/01\
\01/
·x2

POL(f(x1, x2)) =
/0\
\0/
 +
/00\
\10/
·x1 +
/00\
\01/
·x2

POL(a) =
/1\
\0/

POL(h(x1)) =
/0\
\1/
 +
/00\
\00/
·x1

The following usable rules [FROCOS05] were oriented:

f(x, f(a, y)) → f(a, f(f(f(a, x), h(a)), y))

(6) Obligation:

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

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

The TRS R consists of the following rules:

f(x, f(a, y)) → f(a, f(f(f(a, x), h(a)), y))

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

(7) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • F(x, f(a, y)) → F(f(f(a, x), h(a)), y)
    The graph contains the following edges 2 > 2

(8) TRUE