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:

le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, x) -> 0
minus2(x, 0) -> x
minus2(0, x) -> 0
minus2(s1(x), s1(y)) -> minus2(x, y)
isZero1(0) -> true
isZero1(s1(x)) -> false
mod2(x, y) -> if_mod5(isZero1(y), le2(y, x), x, y, minus2(x, y))
if_mod5(true, b, x, y, z) -> divByZeroError
if_mod5(false, false, x, y, z) -> x
if_mod5(false, true, x, y, z) -> mod2(z, y)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, x) -> 0
minus2(x, 0) -> x
minus2(0, x) -> 0
minus2(s1(x), s1(y)) -> minus2(x, y)
isZero1(0) -> true
isZero1(s1(x)) -> false
mod2(x, y) -> if_mod5(isZero1(y), le2(y, x), x, y, minus2(x, y))
if_mod5(true, b, x, y, z) -> divByZeroError
if_mod5(false, false, x, y, z) -> x
if_mod5(false, true, x, y, z) -> mod2(z, 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:

MOD2(x, y) -> MINUS2(x, y)
LE2(s1(x), s1(y)) -> LE2(x, y)
MOD2(x, y) -> ISZERO1(y)
MOD2(x, y) -> IF_MOD5(isZero1(y), le2(y, x), x, y, minus2(x, y))
MINUS2(s1(x), s1(y)) -> MINUS2(x, y)
IF_MOD5(false, true, x, y, z) -> MOD2(z, y)
MOD2(x, y) -> LE2(y, x)

The TRS R consists of the following rules:

le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, x) -> 0
minus2(x, 0) -> x
minus2(0, x) -> 0
minus2(s1(x), s1(y)) -> minus2(x, y)
isZero1(0) -> true
isZero1(s1(x)) -> false
mod2(x, y) -> if_mod5(isZero1(y), le2(y, x), x, y, minus2(x, y))
if_mod5(true, b, x, y, z) -> divByZeroError
if_mod5(false, false, x, y, z) -> x
if_mod5(false, true, x, y, z) -> mod2(z, 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:

MOD2(x, y) -> MINUS2(x, y)
LE2(s1(x), s1(y)) -> LE2(x, y)
MOD2(x, y) -> ISZERO1(y)
MOD2(x, y) -> IF_MOD5(isZero1(y), le2(y, x), x, y, minus2(x, y))
MINUS2(s1(x), s1(y)) -> MINUS2(x, y)
IF_MOD5(false, true, x, y, z) -> MOD2(z, y)
MOD2(x, y) -> LE2(y, x)

The TRS R consists of the following rules:

le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, x) -> 0
minus2(x, 0) -> x
minus2(0, x) -> 0
minus2(s1(x), s1(y)) -> minus2(x, y)
isZero1(0) -> true
isZero1(s1(x)) -> false
mod2(x, y) -> if_mod5(isZero1(y), le2(y, x), x, y, minus2(x, y))
if_mod5(true, b, x, y, z) -> divByZeroError
if_mod5(false, false, x, y, z) -> x
if_mod5(false, true, x, y, z) -> mod2(z, y)

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 3 less nodes.

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

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

MINUS2(s1(x), s1(y)) -> MINUS2(x, y)

The TRS R consists of the following rules:

le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, x) -> 0
minus2(x, 0) -> x
minus2(0, x) -> 0
minus2(s1(x), s1(y)) -> minus2(x, y)
isZero1(0) -> true
isZero1(s1(x)) -> false
mod2(x, y) -> if_mod5(isZero1(y), le2(y, x), x, y, minus2(x, y))
if_mod5(true, b, x, y, z) -> divByZeroError
if_mod5(false, false, x, y, z) -> x
if_mod5(false, true, x, y, z) -> mod2(z, 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.


MINUS2(s1(x), s1(y)) -> MINUS2(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( MINUS2(x1, x2) ) = x2


POL( s1(x1) ) = x1 + 1



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:

le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, x) -> 0
minus2(x, 0) -> x
minus2(0, x) -> 0
minus2(s1(x), s1(y)) -> minus2(x, y)
isZero1(0) -> true
isZero1(s1(x)) -> false
mod2(x, y) -> if_mod5(isZero1(y), le2(y, x), x, y, minus2(x, y))
if_mod5(true, b, x, y, z) -> divByZeroError
if_mod5(false, false, x, y, z) -> x
if_mod5(false, true, x, y, z) -> mod2(z, 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
          ↳ QDP

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

LE2(s1(x), s1(y)) -> LE2(x, y)

The TRS R consists of the following rules:

le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, x) -> 0
minus2(x, 0) -> x
minus2(0, x) -> 0
minus2(s1(x), s1(y)) -> minus2(x, y)
isZero1(0) -> true
isZero1(s1(x)) -> false
mod2(x, y) -> if_mod5(isZero1(y), le2(y, x), x, y, minus2(x, y))
if_mod5(true, b, x, y, z) -> divByZeroError
if_mod5(false, false, x, y, z) -> x
if_mod5(false, true, x, y, z) -> mod2(z, 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.


LE2(s1(x), s1(y)) -> LE2(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( LE2(x1, x2) ) = x2


POL( s1(x1) ) = x1 + 1



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:

le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, x) -> 0
minus2(x, 0) -> x
minus2(0, x) -> 0
minus2(s1(x), s1(y)) -> minus2(x, y)
isZero1(0) -> true
isZero1(s1(x)) -> false
mod2(x, y) -> if_mod5(isZero1(y), le2(y, x), x, y, minus2(x, y))
if_mod5(true, b, x, y, z) -> divByZeroError
if_mod5(false, false, x, y, z) -> x
if_mod5(false, true, x, y, z) -> mod2(z, 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

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

MOD2(x, y) -> IF_MOD5(isZero1(y), le2(y, x), x, y, minus2(x, y))
IF_MOD5(false, true, x, y, z) -> MOD2(z, y)

The TRS R consists of the following rules:

le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, x) -> 0
minus2(x, 0) -> x
minus2(0, x) -> 0
minus2(s1(x), s1(y)) -> minus2(x, y)
isZero1(0) -> true
isZero1(s1(x)) -> false
mod2(x, y) -> if_mod5(isZero1(y), le2(y, x), x, y, minus2(x, y))
if_mod5(true, b, x, y, z) -> divByZeroError
if_mod5(false, false, x, y, z) -> x
if_mod5(false, true, x, y, z) -> mod2(z, y)

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