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

0 QTRS
  ↳1 QTRS Reverse (⇔)
    ↳2 QTRS
  ↳3 QTRS Reverse (⇔)
    ↳4 QTRS
  ↳5 Overlay + Local Confluence (⇔)
    ↳6 QTRS
    ↳7 DependencyPairsProof (⇔)
    ↳8 QDP
      ↳9 MNOCProof (⇔)
        ↳10 QDP
      ↳11 QDPOrderProof (⇔)
        ↳12 QDP
        ↳13 QDPOrderProof (⇔)
        ↳14 QDP
        ↳15 QDPOrderProof (⇔)
        ↳16 QDP
        ↳17 DependencyGraphProof (⇔)
        ↳18 TRUE

(0) Obligation:

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

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

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

d(a(x)) → a(b(c(d(x))))
c(b(x)) → b(a(d(c(x))))
c(a(x)) → x
d(b(x)) → x

Q is empty.

(3) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(4) Obligation:

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

d(a(x)) → a(b(c(d(x))))
c(b(x)) → b(a(d(c(x))))
c(a(x)) → x
d(b(x)) → x

Q is empty.

(5) Overlay + Local Confluence (EQUIVALENT transformation)

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

(6) Obligation:

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

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

The set Q consists of the following terms:

a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))

(7) DependencyPairsProof (EQUIVALENT transformation)

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

(8) Obligation:

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

A(d(x)) → B(a(x))
A(d(x)) → A(x)
B(c(x)) → A(b(x))
B(c(x)) → B(x)

The TRS R consists of the following rules:

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

The set Q consists of the following terms:

a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))

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:

A(d(x)) → B(a(x))
A(d(x)) → A(x)
B(c(x)) → A(b(x))
B(c(x)) → B(x)

The TRS R consists of the following rules:

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

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.


B(c(x)) → B(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A(x1)) =
/0A\
\-I/
 +
/0A0A\
\-I0A/
·x1

POL(d(x1)) =
/1A\
\0A/
 +
/1A0A\
\0A0A/
·x1

POL(B(x1)) =
/0A\
\0A/
 +
/-I0A\
\-I0A/
·x1

POL(a(x1)) =
/0A\
\0A/
 +
/0A-I\
\0A-I/
·x1

POL(c(x1)) =
/-I\
\1A/
 +
/0A0A\
\0A1A/
·x1

POL(b(x1)) =
/0A\
\0A/
 +
/-I0A\
\0A0A/
·x1

The following usable rules [FROCOS05] were oriented:

b(c(x)) → c(d(a(b(x))))
a(d(x)) → d(c(b(a(x))))
b(d(x)) → x
a(c(x)) → x

(12) Obligation:

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

A(d(x)) → B(a(x))
A(d(x)) → A(x)
B(c(x)) → A(b(x))

The TRS R consists of the following rules:

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

The set Q consists of the following terms:

a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A(d(x)) → A(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A(x1)) =
/0A\
\-I/
 +
/-I0A\
\-I-I/
·x1

POL(d(x1)) =
/0A\
\1A/
 +
/-I0A\
\0A1A/
·x1

POL(B(x1)) =
/1A\
\-I/
 +
/-I0A\
\-I-I/
·x1

POL(a(x1)) =
/0A\
\0A/
 +
/-I0A\
\0A1A/
·x1

POL(c(x1)) =
/0A\
\-I/
 +
/1A0A\
\0A-I/
·x1

POL(b(x1)) =
/0A\
\-I/
 +
/1A0A\
\0A-I/
·x1

The following usable rules [FROCOS05] were oriented:

b(c(x)) → c(d(a(b(x))))
a(d(x)) → d(c(b(a(x))))
b(d(x)) → x
a(c(x)) → x

(14) Obligation:

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

A(d(x)) → B(a(x))
B(c(x)) → A(b(x))

The TRS R consists of the following rules:

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

The set Q consists of the following terms:

a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A(d(x)) → B(a(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A(x1)) =
/1A\
\-I/
 +
/0A1A\
\-I-I/
·x1

POL(d(x1)) =
/0A\
\1A/
 +
/-I0A\
\0A1A/
·x1

POL(B(x1)) =
/1A\
\-I/
 +
/0A0A\
\-I-I/
·x1

POL(a(x1)) =
/0A\
\-I/
 +
/-I0A\
\0A1A/
·x1

POL(c(x1)) =
/0A\
\0A/
 +
/1A0A\
\0A-I/
·x1

POL(b(x1)) =
/-I\
\-I/
 +
/1A0A\
\0A-I/
·x1

The following usable rules [FROCOS05] were oriented:

b(c(x)) → c(d(a(b(x))))
a(d(x)) → d(c(b(a(x))))
b(d(x)) → x
a(c(x)) → x

(16) Obligation:

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

B(c(x)) → A(b(x))

The TRS R consists of the following rules:

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

The set Q consists of the following terms:

a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(18) TRUE