Termination w.r.t. Q of the following Term Rewriting System could be proven:

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.


↳ QTRS
  ↳ Overlay + Local Confluence

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.

The TRS is overlay and locally confluent. By [19] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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))


Using Dependency Pairs [1,15] we result in the following initial DP problem:
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.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ Narrowing

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.
By narrowing [15] the rule H(f(x, y)) → H(h(x)) at position [0] we obtained the following new rules:

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



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ ForwardInstantiation

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(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.
By forward instantiating [14] the rule H(f(x, y)) → H(x) we obtained the following new rules:

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



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ ForwardInstantiation
                ↳ QDP
                  ↳ MNOCProof
                  ↳ QDPOrderProof

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

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.
We use the modular non-overlap check [17] to decrease Q to the empty set.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ ForwardInstantiation
                ↳ QDP
                  ↳ MNOCProof
                    ↳ QDP
                  ↳ QDPOrderProof

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

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.
We use the reduction pair processor [15].


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(f(y_0, y_1), y_2), x1)) → H(f(f(y_0, y_1), y_2))
H(f(f(y_0, y_1), x1)) → H(f(y_0, y_1))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( a ) =
/0\
\0/

M( f(x1, x2) ) =
/2\
\1/
+
/03\
\01/
·x1+
/01\
\01/
·x2

M( h(x1) ) =
/0\
\0/
+
/03\
\10/
·x1

Tuple symbols:
M( H(x1) ) = 0+
[3,0]
·x1


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

h(f(x, y)) → f(y, f(h(h(x)), a))



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ ForwardInstantiation
                ↳ QDP
                  ↳ MNOCProof
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ PisEmptyProof

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