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

0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 Narrowing (⇔)
↳6 QDP
↳7 ForwardInstantiation (⇔)
↳8 QDP
  ↳9 MNOCProof (⇔)
    ↳10 QDP
  ↳11 QDPOrderProof (⇔)
    ↳12 QDP
    ↳13 PisEmptyProof (⇔)
    ↳14 TRUE

(0) Obligation:

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

h(f(x, y)) → f(y, f(h(h(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:

h(f(x, y)) → f(y, f(h(h(x)), a))

The set Q consists of the following terms:

h(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:

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

The TRS R consists of the following rules:

h(f(x, y)) → f(y, f(h(h(x)), a))

The set Q consists of the following terms:

h(f(x0, x1))

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

(5) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule H(f(x, y)) → H(h(x)) at position [0] we obtained the following new rules [LPAR04]:

H(f(f(x0, x1), y1)) → H(f(x1, f(h(h(x0)), a)))

(6) Obligation:

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

H(f(x, y)) → H(x)
H(f(f(x0, x1), y1)) → H(f(x1, f(h(h(x0)), a)))

The TRS R consists of the following rules:

h(f(x, y)) → f(y, f(h(h(x)), a))

The set Q consists of the following terms:

h(f(x0, x1))

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

(7) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule H(f(x, y)) → H(x) we obtained the following new rules [LPAR04]:

H(f(f(y_0, y_1), x1)) → H(f(y_0, y_1))
H(f(f(f(y_0, y_1), y_2), x1)) → H(f(f(y_0, y_1), y_2))

(8) Obligation:

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

H(f(f(x0, x1), y1)) → H(f(x1, f(h(h(x0)), a)))
H(f(f(y_0, y_1), x1)) → H(f(y_0, y_1))
H(f(f(f(y_0, y_1), y_2), x1)) → H(f(f(y_0, y_1), y_2))

The TRS R consists of the following rules:

h(f(x, y)) → f(y, f(h(h(x)), a))

The set Q consists of the following terms:

h(f(x0, x1))

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

(9) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.

(10) Obligation:

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

H(f(f(x0, x1), y1)) → H(f(x1, f(h(h(x0)), a)))
H(f(f(y_0, y_1), x1)) → H(f(y_0, y_1))
H(f(f(f(y_0, y_1), y_2), x1)) → H(f(f(y_0, y_1), y_2))

The TRS R consists of the following rules:

h(f(x, y)) → f(y, f(h(h(x)), a))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


H(f(f(x0, x1), y1)) → H(f(x1, f(h(h(x0)), a)))
H(f(f(y_0, y_1), x1)) → H(f(y_0, y_1))
H(f(f(f(y_0, y_1), y_2), x1)) → H(f(f(y_0, y_1), y_2))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(H(x1)) =
/0\
\0/
 +
/01\
\00/
·x1

POL(f(x1, x2)) =
/1\
\2/
 +
/10\
\30/
·x1 +
/10\
\10/
·x2

POL(h(x1)) =
/0\
\0/
 +
/01\
\30/
·x1

POL(a) =
/0\
\0/

The following usable rules [FROCOS05] were oriented:

h(f(x, y)) → f(y, f(h(h(x)), a))

(12) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

h(f(x, y)) → f(y, f(h(h(x)), a))

The set Q consists of the following terms:

h(f(x0, x1))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE