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:

a(d(x1)) → d(b(x1))
a(x1) → b(b(b(x1)))
d(x1) → x1
a(x1) → x1
b(d(b(x1))) → a(d(x1))
b(c(x1)) → c(d(d(x1)))
a(c(x1)) → b(b(c(d(x1))))

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

a(d(x1)) → d(b(x1))
a(x1) → b(b(b(x1)))
d(x1) → x1
a(x1) → x1
b(d(b(x1))) → a(d(x1))
b(c(x1)) → c(d(d(x1)))
a(c(x1)) → b(b(c(d(x1))))

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

A(d(x1)) → B(x1)
A(x1) → B(b(x1))
A(c(x1)) → B(c(d(x1)))
B(d(b(x1))) → A(d(x1))
A(x1) → B(b(b(x1)))
B(d(b(x1))) → D(x1)
A(d(x1)) → D(b(x1))
A(x1) → B(x1)
A(c(x1)) → B(b(c(d(x1))))
B(c(x1)) → D(d(x1))
A(c(x1)) → D(x1)
B(c(x1)) → D(x1)

The TRS R consists of the following rules:

a(d(x1)) → d(b(x1))
a(x1) → b(b(b(x1)))
d(x1) → x1
a(x1) → x1
b(d(b(x1))) → a(d(x1))
b(c(x1)) → c(d(d(x1)))
a(c(x1)) → b(b(c(d(x1))))

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

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

A(d(x1)) → B(x1)
A(x1) → B(b(x1))
A(c(x1)) → B(c(d(x1)))
B(d(b(x1))) → A(d(x1))
A(x1) → B(b(b(x1)))
B(d(b(x1))) → D(x1)
A(d(x1)) → D(b(x1))
A(x1) → B(x1)
A(c(x1)) → B(b(c(d(x1))))
B(c(x1)) → D(d(x1))
A(c(x1)) → D(x1)
B(c(x1)) → D(x1)

The TRS R consists of the following rules:

a(d(x1)) → d(b(x1))
a(x1) → b(b(b(x1)))
d(x1) → x1
a(x1) → x1
b(d(b(x1))) → a(d(x1))
b(c(x1)) → c(d(d(x1)))
a(c(x1)) → b(b(c(d(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP
          ↳ QDPOrderProof

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

A(d(x1)) → B(x1)
A(x1) → B(b(x1))
A(x1) → B(b(b(x1)))
B(d(b(x1))) → A(d(x1))
A(x1) → B(x1)
A(c(x1)) → B(b(c(d(x1))))

The TRS R consists of the following rules:

a(d(x1)) → d(b(x1))
a(x1) → b(b(b(x1)))
d(x1) → x1
a(x1) → x1
b(d(b(x1))) → a(d(x1))
b(c(x1)) → c(d(d(x1)))
a(c(x1)) → b(b(c(d(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A(d(x1)) → B(x1)
The remaining pairs can at least be oriented weakly.

A(x1) → B(b(x1))
A(x1) → B(b(b(x1)))
B(d(b(x1))) → A(d(x1))
A(x1) → B(x1)
A(c(x1)) → B(b(c(d(x1))))
Used ordering: Polynomial Order [21,25] with Interpretation:

POL( A(x1) ) = x1


POL( c(x1) ) = max{0, -1}


POL( b(x1) ) = x1


POL( d(x1) ) = x1 + 1


POL( B(x1) ) = x1


POL( a(x1) ) = x1



The following usable rules [17] were oriented:

b(c(x1)) → c(d(d(x1)))
a(d(x1)) → d(b(x1))
a(x1) → x1
b(d(b(x1))) → a(d(x1))
a(c(x1)) → b(b(c(d(x1))))
a(x1) → b(b(b(x1)))
d(x1) → x1



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
QDP
              ↳ QDPOrderProof

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

A(x1) → B(b(x1))
B(d(b(x1))) → A(d(x1))
A(x1) → B(b(b(x1)))
A(x1) → B(x1)
A(c(x1)) → B(b(c(d(x1))))

The TRS R consists of the following rules:

a(d(x1)) → d(b(x1))
a(x1) → b(b(b(x1)))
d(x1) → x1
a(x1) → x1
b(d(b(x1))) → a(d(x1))
b(c(x1)) → c(d(d(x1)))
a(c(x1)) → b(b(c(d(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A(x1) → B(b(x1))
B(d(b(x1))) → A(d(x1))
A(x1) → B(x1)
A(c(x1)) → B(b(c(d(x1))))
The remaining pairs can at least be oriented weakly.

A(x1) → B(b(b(x1)))
Used ordering: Polynomial interpretation [25]:

POL(A(x1)) = 4 + x1   
POL(B(x1)) = x1   
POL(a(x1)) = 8 + x1   
POL(b(x1)) = 2 + x1   
POL(c(x1)) = 0   
POL(d(x1)) = 2 + 3·x1   

The following usable rules [17] were oriented:

b(c(x1)) → c(d(d(x1)))
a(d(x1)) → d(b(x1))
a(x1) → x1
b(d(b(x1))) → a(d(x1))
a(c(x1)) → b(b(c(d(x1))))
a(x1) → b(b(b(x1)))
d(x1) → x1



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ QDPOrderProof
QDP
                  ↳ DependencyGraphProof

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

A(x1) → B(b(b(x1)))

The TRS R consists of the following rules:

a(d(x1)) → d(b(x1))
a(x1) → b(b(b(x1)))
d(x1) → x1
a(x1) → x1
b(d(b(x1))) → a(d(x1))
b(c(x1)) → c(d(d(x1)))
a(c(x1)) → b(b(c(d(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.