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:

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

Q is empty.

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

F2(s1(x), s1(y)) -> TWICE1(min2(x, y))
MIN2(s1(x), s1(y)) -> MIN2(x, y)
F2(s1(x), s1(y)) -> MIN2(x, y)
-12(s1(x), s1(y)) -> -12(x, y)
F2(s1(x), s1(y)) -> -12(x, min2(x, y))
F2(s1(x), s1(y)) -> F2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))
F2(s1(x), s1(y)) -> F2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
TWICE1(s1(x)) -> TWICE1(x)
F2(s1(x), s1(y)) -> -12(y, min2(x, y))

The TRS R consists of the following rules:

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

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:

F2(s1(x), s1(y)) -> TWICE1(min2(x, y))
MIN2(s1(x), s1(y)) -> MIN2(x, y)
F2(s1(x), s1(y)) -> MIN2(x, y)
-12(s1(x), s1(y)) -> -12(x, y)
F2(s1(x), s1(y)) -> -12(x, min2(x, y))
F2(s1(x), s1(y)) -> F2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))
F2(s1(x), s1(y)) -> F2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
TWICE1(s1(x)) -> TWICE1(x)
F2(s1(x), s1(y)) -> -12(y, min2(x, y))

The TRS R consists of the following rules:

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
QDP
            ↳ QDPAfsSolverProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

TWICE1(s1(x)) -> TWICE1(x)

The TRS R consists of the following rules:

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

TWICE1(s1(x)) -> TWICE1(x)
Used argument filtering: TWICE1(x1)  =  x1
s1(x1)  =  s1(x1)
Used ordering: Quasi Precedence: trivial


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPAfsSolverProof
QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

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
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
QDP
            ↳ QDPAfsSolverProof
          ↳ QDP
          ↳ QDP

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

MIN2(s1(x), s1(y)) -> MIN2(x, y)

The TRS R consists of the following rules:

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

MIN2(s1(x), s1(y)) -> MIN2(x, y)
Used argument filtering: MIN2(x1, x2)  =  x2
s1(x1)  =  s1(x1)
Used ordering: Quasi Precedence: trivial


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPAfsSolverProof
QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

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

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

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
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPAfsSolverProof
          ↳ QDP

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

-12(s1(x), s1(y)) -> -12(x, y)

The TRS R consists of the following rules:

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

-12(s1(x), s1(y)) -> -12(x, y)
Used argument filtering: -12(x1, x2)  =  x2
s1(x1)  =  s1(x1)
Used ordering: Quasi Precedence: trivial


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPAfsSolverProof
QDP
                ↳ PisEmptyProof
          ↳ QDP

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

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

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
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP

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

F2(s1(x), s1(y)) -> F2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
F2(s1(x), s1(y)) -> F2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

The TRS R consists of the following rules:

-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
min2(x, 0) -> 0
min2(0, y) -> 0
min2(s1(x), s1(y)) -> s1(min2(x, y))
twice1(0) -> 0
twice1(s1(x)) -> s1(s1(twice1(x)))
f2(s1(x), s1(y)) -> f2(-2(y, min2(x, y)), s1(twice1(min2(x, y))))
f2(s1(x), s1(y)) -> f2(-2(x, min2(x, y)), s1(twice1(min2(x, y))))

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