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:

not1(true) -> false
not1(false) -> true
odd1(0) -> false
odd1(s1(x)) -> not1(odd1(x))
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
+2(s1(x), y) -> s1(+2(x, y))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

not1(true) -> false
not1(false) -> true
odd1(0) -> false
odd1(s1(x)) -> not1(odd1(x))
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
+2(s1(x), y) -> s1(+2(x, y))

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:

ODD1(s1(x)) -> NOT1(odd1(x))
+12(x, s1(y)) -> +12(x, y)
+12(s1(x), y) -> +12(x, y)
ODD1(s1(x)) -> ODD1(x)

The TRS R consists of the following rules:

not1(true) -> false
not1(false) -> true
odd1(0) -> false
odd1(s1(x)) -> not1(odd1(x))
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
+2(s1(x), y) -> s1(+2(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:

ODD1(s1(x)) -> NOT1(odd1(x))
+12(x, s1(y)) -> +12(x, y)
+12(s1(x), y) -> +12(x, y)
ODD1(s1(x)) -> ODD1(x)

The TRS R consists of the following rules:

not1(true) -> false
not1(false) -> true
odd1(0) -> false
odd1(s1(x)) -> not1(odd1(x))
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
+2(s1(x), y) -> s1(+2(x, y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 1 less node.

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

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

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

The TRS R consists of the following rules:

not1(true) -> false
not1(false) -> true
odd1(0) -> false
odd1(s1(x)) -> not1(odd1(x))
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
+2(s1(x), y) -> s1(+2(x, y))

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.


+12(x, s1(y)) -> +12(x, y)
The remaining pairs can at least be oriented weakly.

+12(s1(x), y) -> +12(x, y)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( +12(x1, x2) ) = max{0, x2 - 1}


POL( s1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



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

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

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

The TRS R consists of the following rules:

not1(true) -> false
not1(false) -> true
odd1(0) -> false
odd1(s1(x)) -> not1(odd1(x))
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
+2(s1(x), y) -> s1(+2(x, y))

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.


+12(s1(x), y) -> +12(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( +12(x1, x2) ) = max{0, x1 - 1}


POL( s1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



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

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

not1(true) -> false
not1(false) -> true
odd1(0) -> false
odd1(s1(x)) -> not1(odd1(x))
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
+2(s1(x), y) -> s1(+2(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
            ↳ QDPOrderProof

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

ODD1(s1(x)) -> ODD1(x)

The TRS R consists of the following rules:

not1(true) -> false
not1(false) -> true
odd1(0) -> false
odd1(s1(x)) -> not1(odd1(x))
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
+2(s1(x), y) -> s1(+2(x, y))

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.


ODD1(s1(x)) -> ODD1(x)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ODD1(x1) ) = max{0, x1 - 1}


POL( s1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



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

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

not1(true) -> false
not1(false) -> true
odd1(0) -> false
odd1(s1(x)) -> not1(odd1(x))
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
+2(s1(x), y) -> s1(+2(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.