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:

p1(0) -> 0
p1(s1(x)) -> x
le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, y) -> if3(le2(x, y), x, y)
if3(true, x, y) -> 0
if3(false, x, y) -> s1(minus2(p1(x), y))

Q is empty.


QTRS
  ↳ Non-Overlap Check

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

p1(0) -> 0
p1(s1(x)) -> x
le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, y) -> if3(le2(x, y), x, y)
if3(true, x, y) -> 0
if3(false, x, y) -> s1(minus2(p1(x), y))

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS
  ↳ Non-Overlap Check
QTRS
      ↳ DependencyPairsProof

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

p1(0) -> 0
p1(s1(x)) -> x
le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, y) -> if3(le2(x, y), x, y)
if3(true, x, y) -> 0
if3(false, x, y) -> s1(minus2(p1(x), y))

The set Q consists of the following terms:

p1(0)
p1(s1(x0))
le2(0, x0)
le2(s1(x0), 0)
le2(s1(x0), s1(x1))
minus2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)


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

MINUS2(x, y) -> LE2(x, y)
MINUS2(x, y) -> IF3(le2(x, y), x, y)
LE2(s1(x), s1(y)) -> LE2(x, y)
IF3(false, x, y) -> MINUS2(p1(x), y)
IF3(false, x, y) -> P1(x)

The TRS R consists of the following rules:

p1(0) -> 0
p1(s1(x)) -> x
le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, y) -> if3(le2(x, y), x, y)
if3(true, x, y) -> 0
if3(false, x, y) -> s1(minus2(p1(x), y))

The set Q consists of the following terms:

p1(0)
p1(s1(x0))
le2(0, x0)
le2(s1(x0), 0)
le2(s1(x0), s1(x1))
minus2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
QDP
          ↳ DependencyGraphProof

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

MINUS2(x, y) -> LE2(x, y)
MINUS2(x, y) -> IF3(le2(x, y), x, y)
LE2(s1(x), s1(y)) -> LE2(x, y)
IF3(false, x, y) -> MINUS2(p1(x), y)
IF3(false, x, y) -> P1(x)

The TRS R consists of the following rules:

p1(0) -> 0
p1(s1(x)) -> x
le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, y) -> if3(le2(x, y), x, y)
if3(true, x, y) -> 0
if3(false, x, y) -> s1(minus2(p1(x), y))

The set Q consists of the following terms:

p1(0)
p1(s1(x0))
le2(0, x0)
le2(s1(x0), 0)
le2(s1(x0), s1(x1))
minus2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 2 SCCs with 2 less nodes.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
QDP
                ↳ QDPAfsSolverProof
              ↳ 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:

p1(0) -> 0
p1(s1(x)) -> x
le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, y) -> if3(le2(x, y), x, y)
if3(true, x, y) -> 0
if3(false, x, y) -> s1(minus2(p1(x), y))

The set Q consists of the following terms:

p1(0)
p1(s1(x0))
le2(0, x0)
le2(s1(x0), 0)
le2(s1(x0), s1(x1))
minus2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)

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.

LE2(s1(x), s1(y)) -> LE2(x, y)
Used argument filtering: LE2(x1, x2)  =  x2
s1(x1)  =  s1(x1)
Used ordering: Quasi Precedence: trivial


↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPAfsSolverProof
QDP
                    ↳ PisEmptyProof
              ↳ QDP

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

p1(0) -> 0
p1(s1(x)) -> x
le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, y) -> if3(le2(x, y), x, y)
if3(true, x, y) -> 0
if3(false, x, y) -> s1(minus2(p1(x), y))

The set Q consists of the following terms:

p1(0)
p1(s1(x0))
le2(0, x0)
le2(s1(x0), 0)
le2(s1(x0), s1(x1))
minus2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
QDP

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

MINUS2(x, y) -> IF3(le2(x, y), x, y)
IF3(false, x, y) -> MINUS2(p1(x), y)

The TRS R consists of the following rules:

p1(0) -> 0
p1(s1(x)) -> x
le2(0, y) -> true
le2(s1(x), 0) -> false
le2(s1(x), s1(y)) -> le2(x, y)
minus2(x, y) -> if3(le2(x, y), x, y)
if3(true, x, y) -> 0
if3(false, x, y) -> s1(minus2(p1(x), y))

The set Q consists of the following terms:

p1(0)
p1(s1(x0))
le2(0, x0)
le2(s1(x0), 0)
le2(s1(x0), s1(x1))
minus2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)

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