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:

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, z))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, 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:

DIV2(div2(x, y), z) -> TIMES2(y, z)
PLUS2(s1(x), y) -> PLUS2(x, y)
TIMES2(s1(x), y) -> PLUS2(y, times2(x, y))
QUOT3(x, 0, s1(z)) -> DIV2(x, s1(z))
QUOT3(s1(x), s1(y), z) -> QUOT3(x, y, z)
DIV2(div2(x, y), z) -> DIV2(x, times2(y, z))
TIMES2(s1(x), y) -> TIMES2(x, y)
DIV2(x, y) -> QUOT3(x, y, y)

The TRS R consists of the following rules:

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, z))

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:

DIV2(div2(x, y), z) -> TIMES2(y, z)
PLUS2(s1(x), y) -> PLUS2(x, y)
TIMES2(s1(x), y) -> PLUS2(y, times2(x, y))
QUOT3(x, 0, s1(z)) -> DIV2(x, s1(z))
QUOT3(s1(x), s1(y), z) -> QUOT3(x, y, z)
DIV2(div2(x, y), z) -> DIV2(x, times2(y, z))
TIMES2(s1(x), y) -> TIMES2(x, y)
DIV2(x, y) -> QUOT3(x, y, y)

The TRS R consists of the following rules:

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, z))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 2 less nodes.

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

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

PLUS2(s1(x), y) -> PLUS2(x, y)

The TRS R consists of the following rules:

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, 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.


PLUS2(s1(x), y) -> PLUS2(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PLUS2(x1, x2)) = 2·x1   
POL(s1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



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

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

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, 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
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

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

TIMES2(s1(x), y) -> TIMES2(x, y)

The TRS R consists of the following rules:

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, 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.


TIMES2(s1(x), y) -> TIMES2(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(TIMES2(x1, x2)) = 2·x1   
POL(s1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



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

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

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, 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
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof

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

QUOT3(x, 0, s1(z)) -> DIV2(x, s1(z))
QUOT3(s1(x), s1(y), z) -> QUOT3(x, y, z)
DIV2(div2(x, y), z) -> DIV2(x, times2(y, z))
DIV2(x, y) -> QUOT3(x, y, y)

The TRS R consists of the following rules:

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, 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.


DIV2(div2(x, y), z) -> DIV2(x, times2(y, z))
The remaining pairs can at least be oriented weakly.

QUOT3(x, 0, s1(z)) -> DIV2(x, s1(z))
QUOT3(s1(x), s1(y), z) -> QUOT3(x, y, z)
DIV2(x, y) -> QUOT3(x, y, y)
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(DIV2(x1, x2)) = 2·x1   
POL(QUOT3(x1, x2, x3)) = 2·x1   
POL(div2(x1, x2)) = 2 + 2·x1   
POL(plus2(x1, x2)) = 0   
POL(s1(x1)) = 2·x1   
POL(times2(x1, x2)) = 0   

The following usable rules [14] were oriented: none



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

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

QUOT3(s1(x), s1(y), z) -> QUOT3(x, y, z)
QUOT3(x, 0, s1(z)) -> DIV2(x, s1(z))
DIV2(x, y) -> QUOT3(x, y, y)

The TRS R consists of the following rules:

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, 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.


QUOT3(s1(x), s1(y), z) -> QUOT3(x, y, z)
The remaining pairs can at least be oriented weakly.

QUOT3(x, 0, s1(z)) -> DIV2(x, s1(z))
DIV2(x, y) -> QUOT3(x, y, y)
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(DIV2(x1, x2)) = x1   
POL(QUOT3(x1, x2, x3)) = x1   
POL(s1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



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

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

QUOT3(x, 0, s1(z)) -> DIV2(x, s1(z))
DIV2(x, y) -> QUOT3(x, y, y)

The TRS R consists of the following rules:

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, 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.


QUOT3(x, 0, s1(z)) -> DIV2(x, s1(z))
The remaining pairs can at least be oriented weakly.

DIV2(x, y) -> QUOT3(x, y, y)
Used ordering: Polynomial interpretation [21]:

POL(0) = 2   
POL(DIV2(x1, x2)) = 2·x2   
POL(QUOT3(x1, x2, x3)) = x2   
POL(s1(x1)) = 0   

The following usable rules [14] were oriented: none



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

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

DIV2(x, y) -> QUOT3(x, y, y)

The TRS R consists of the following rules:

plus2(x, 0) -> x
plus2(0, y) -> y
plus2(s1(x), y) -> s1(plus2(x, y))
times2(0, y) -> 0
times2(s1(0), y) -> y
times2(s1(x), y) -> plus2(y, times2(x, y))
div2(0, y) -> 0
div2(x, y) -> quot3(x, y, y)
quot3(0, s1(y), z) -> 0
quot3(s1(x), s1(y), z) -> quot3(x, y, z)
quot3(x, 0, s1(z)) -> s1(div2(x, s1(z)))
div2(div2(x, y), z) -> div2(x, times2(y, z))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.