Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

MIN(s(x), s(y)) → MIN(x, y)
-1(s(x), s(y)) → -1(x, y)
GCD(x, s(y), s(z)) → GCD(x, -(max(y, z), min(y, z)), s(min(y, z)))
GCD(s(x), s(y), z) → -1(max(x, y), min(x, y))
GCD(s(x), y, s(z)) → GCD(-(max(x, z), min(x, z)), y, s(min(x, z)))
GCD(x, s(y), s(z)) → -1(max(y, z), min(y, z))
GCD(s(x), s(y), z) → GCD(-(max(x, y), min(x, y)), s(min(x, y)), z)
GCD(x, s(y), s(z)) → MIN(y, z)
GCD(s(x), y, s(z)) → MAX(x, z)
GCD(x, s(y), s(z)) → MAX(y, z)
GCD(s(x), s(y), z) → MIN(x, y)
GCD(s(x), s(y), z) → MAX(x, y)
MAX(s(x), s(y)) → MAX(x, y)
GCD(s(x), y, s(z)) → MIN(x, z)
GCD(s(x), y, s(z)) → -1(max(x, z), min(x, z))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof

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

MIN(s(x), s(y)) → MIN(x, y)
-1(s(x), s(y)) → -1(x, y)
GCD(x, s(y), s(z)) → GCD(x, -(max(y, z), min(y, z)), s(min(y, z)))
GCD(s(x), s(y), z) → -1(max(x, y), min(x, y))
GCD(s(x), y, s(z)) → GCD(-(max(x, z), min(x, z)), y, s(min(x, z)))
GCD(x, s(y), s(z)) → -1(max(y, z), min(y, z))
GCD(s(x), s(y), z) → GCD(-(max(x, y), min(x, y)), s(min(x, y)), z)
GCD(x, s(y), s(z)) → MIN(y, z)
GCD(s(x), y, s(z)) → MAX(x, z)
GCD(x, s(y), s(z)) → MAX(y, z)
GCD(s(x), s(y), z) → MIN(x, y)
GCD(s(x), s(y), z) → MAX(x, y)
MAX(s(x), s(y)) → MAX(x, y)
GCD(s(x), y, s(z)) → MIN(x, z)
GCD(s(x), y, s(z)) → -1(max(x, z), min(x, z))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof

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

MIN(s(x), s(y)) → MIN(x, y)
GCD(x, s(y), s(z)) → GCD(x, -(max(y, z), min(y, z)), s(min(y, z)))
-1(s(x), s(y)) → -1(x, y)
GCD(s(x), s(y), z) → -1(max(x, y), min(x, y))
GCD(s(x), y, s(z)) → GCD(-(max(x, z), min(x, z)), y, s(min(x, z)))
GCD(x, s(y), s(z)) → -1(max(y, z), min(y, z))
GCD(s(x), s(y), z) → GCD(-(max(x, y), min(x, y)), s(min(x, y)), z)
GCD(x, s(y), s(z)) → MIN(y, z)
GCD(s(x), y, s(z)) → MAX(x, z)
GCD(x, s(y), s(z)) → MAX(y, z)
GCD(s(x), s(y), z) → MIN(x, y)
GCD(s(x), s(y), z) → MAX(x, y)
MAX(s(x), s(y)) → MAX(x, y)
GCD(s(x), y, s(z)) → -1(max(x, z), min(x, z))
GCD(s(x), y, s(z)) → MIN(x, z)

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 4 SCCs with 9 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

-1(s(x), s(y)) → -1(x, y)

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


-1(s(x), s(y)) → -1(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
-1(x1, x2)  =  x2
s(x1)  =  s(x1)

Recursive Path Order [2].
Precedence:
trivial

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
              ↳ QDP
              ↳ QDP

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

MAX(s(x), s(y)) → MAX(x, y)

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


MAX(s(x), s(y)) → MAX(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
MAX(x1, x2)  =  x2
s(x1)  =  s(x1)

Recursive Path Order [2].
Precedence:
trivial

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP

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

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
              ↳ QDP

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

MIN(s(x), s(y)) → MIN(x, y)

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


MIN(s(x), s(y)) → MIN(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
MIN(x1, x2)  =  x2
s(x1)  =  s(x1)

Recursive Path Order [2].
Precedence:
trivial

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
              ↳ QDP

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

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

GCD(x, s(y), s(z)) → GCD(x, -(max(y, z), min(y, z)), s(min(y, z)))
GCD(s(x), y, s(z)) → GCD(-(max(x, z), min(x, z)), y, s(min(x, z)))
GCD(s(x), s(y), z) → GCD(-(max(x, y), min(x, y)), s(min(x, y)), z)

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

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