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:

app2(D, t) -> 1
app2(D, constant) -> 0
app2(D, app2(app2(+, x), y)) -> app2(app2(+, app2(D, x)), app2(D, y))
app2(D, app2(app2(*, x), y)) -> app2(app2(+, app2(app2(*, y), app2(D, x))), app2(app2(*, x), app2(D, y)))
app2(D, app2(app2(-, x), y)) -> app2(app2(-, app2(D, x)), app2(D, y))
app2(D, app2(minus, x)) -> app2(minus, app2(D, x))
app2(D, app2(app2(div, x), y)) -> app2(app2(-, app2(app2(div, app2(D, x)), y)), app2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2)))
app2(D, app2(ln, x)) -> app2(app2(div, app2(D, x)), x)
app2(D, app2(app2(pow, x), y)) -> app2(app2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))), app2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y)))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

app2(D, t) -> 1
app2(D, constant) -> 0
app2(D, app2(app2(+, x), y)) -> app2(app2(+, app2(D, x)), app2(D, y))
app2(D, app2(app2(*, x), y)) -> app2(app2(+, app2(app2(*, y), app2(D, x))), app2(app2(*, x), app2(D, y)))
app2(D, app2(app2(-, x), y)) -> app2(app2(-, app2(D, x)), app2(D, y))
app2(D, app2(minus, x)) -> app2(minus, app2(D, x))
app2(D, app2(app2(div, x), y)) -> app2(app2(-, app2(app2(div, app2(D, x)), y)), app2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2)))
app2(D, app2(ln, x)) -> app2(app2(div, app2(D, x)), x)
app2(D, app2(app2(pow, x), y)) -> app2(app2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))), app2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, 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:

app2(D, t) -> 1
app2(D, constant) -> 0
app2(D, app2(app2(+, x), y)) -> app2(app2(+, app2(D, x)), app2(D, y))
app2(D, app2(app2(*, x), y)) -> app2(app2(+, app2(app2(*, y), app2(D, x))), app2(app2(*, x), app2(D, y)))
app2(D, app2(app2(-, x), y)) -> app2(app2(-, app2(D, x)), app2(D, y))
app2(D, app2(minus, x)) -> app2(minus, app2(D, x))
app2(D, app2(app2(div, x), y)) -> app2(app2(-, app2(app2(div, app2(D, x)), y)), app2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2)))
app2(D, app2(ln, x)) -> app2(app2(div, app2(D, x)), x)
app2(D, app2(app2(pow, x), y)) -> app2(app2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))), app2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y)))

The set Q consists of the following terms:

app2(D, t)
app2(D, constant)
app2(D, app2(app2(+, x0), x1))
app2(D, app2(app2(*, x0), x1))
app2(D, app2(app2(-, x0), x1))
app2(D, app2(minus, x0))
app2(D, app2(app2(div, x0), x1))
app2(D, app2(ln, x0))
app2(D, app2(app2(pow, x0), x1))


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

APP2(D, app2(app2(-, x), y)) -> APP2(D, y)
APP2(D, app2(minus, x)) -> APP2(minus, app2(D, x))
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))
APP2(D, app2(app2(*, x), y)) -> APP2(D, x)
APP2(D, app2(app2(div, x), y)) -> APP2(app2(div, app2(D, x)), y)
APP2(D, app2(app2(pow, x), y)) -> APP2(ln, x)
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(*, app2(app2(pow, x), y)), app2(ln, x))
APP2(D, app2(app2(div, x), y)) -> APP2(app2(*, x), app2(D, y))
APP2(D, app2(app2(+, x), y)) -> APP2(+, app2(D, x))
APP2(D, app2(app2(div, x), y)) -> APP2(pow, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))), app2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y)))
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(pow, x), app2(app2(-, y), 1))
APP2(D, app2(app2(-, x), y)) -> APP2(-, app2(D, x))
APP2(D, app2(app2(pow, x), y)) -> APP2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x)))
APP2(D, app2(app2(pow, x), y)) -> APP2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x)))
APP2(D, app2(app2(*, x), y)) -> APP2(+, app2(app2(*, y), app2(D, x)))
APP2(D, app2(app2(div, x), y)) -> APP2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2))
APP2(D, app2(app2(div, x), y)) -> APP2(div, app2(app2(*, x), app2(D, y)))
APP2(D, app2(minus, x)) -> APP2(D, x)
APP2(D, app2(app2(div, x), y)) -> APP2(-, app2(app2(div, app2(D, x)), y))
APP2(D, app2(app2(div, x), y)) -> APP2(div, app2(D, x))
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))
APP2(D, app2(ln, x)) -> APP2(div, app2(D, x))
APP2(D, app2(ln, x)) -> APP2(app2(div, app2(D, x)), x)
APP2(D, app2(app2(div, x), y)) -> APP2(D, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(D, x)
APP2(D, app2(app2(+, x), y)) -> APP2(D, x)
APP2(D, app2(app2(-, x), y)) -> APP2(app2(-, app2(D, x)), app2(D, y))
APP2(D, app2(app2(div, x), y)) -> APP2(app2(pow, y), 2)
APP2(D, app2(app2(div, x), y)) -> APP2(*, x)
APP2(D, app2(app2(*, x), y)) -> APP2(D, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(D, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y))
APP2(D, app2(ln, x)) -> APP2(D, x)
APP2(D, app2(app2(pow, x), y)) -> APP2(*, app2(app2(pow, x), y))
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(-, y), 1)
APP2(D, app2(app2(*, x), y)) -> APP2(app2(+, app2(app2(*, y), app2(D, x))), app2(app2(*, x), app2(D, y)))
APP2(D, app2(app2(-, x), y)) -> APP2(D, x)
APP2(D, app2(app2(*, x), y)) -> APP2(app2(*, x), app2(D, y))
APP2(D, app2(app2(*, x), y)) -> APP2(app2(*, y), app2(D, x))
APP2(D, app2(app2(pow, x), y)) -> APP2(*, y)
APP2(D, app2(app2(div, x), y)) -> APP2(app2(-, app2(app2(div, app2(D, x)), y)), app2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2)))
APP2(D, app2(app2(*, x), y)) -> APP2(*, y)
APP2(D, app2(app2(div, x), y)) -> APP2(D, x)
APP2(D, app2(app2(pow, x), y)) -> APP2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1))))
APP2(D, app2(app2(pow, x), y)) -> APP2(-, y)
APP2(D, app2(app2(+, x), y)) -> APP2(D, y)
APP2(D, app2(app2(+, x), y)) -> APP2(app2(+, app2(D, x)), app2(D, y))

The TRS R consists of the following rules:

app2(D, t) -> 1
app2(D, constant) -> 0
app2(D, app2(app2(+, x), y)) -> app2(app2(+, app2(D, x)), app2(D, y))
app2(D, app2(app2(*, x), y)) -> app2(app2(+, app2(app2(*, y), app2(D, x))), app2(app2(*, x), app2(D, y)))
app2(D, app2(app2(-, x), y)) -> app2(app2(-, app2(D, x)), app2(D, y))
app2(D, app2(minus, x)) -> app2(minus, app2(D, x))
app2(D, app2(app2(div, x), y)) -> app2(app2(-, app2(app2(div, app2(D, x)), y)), app2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2)))
app2(D, app2(ln, x)) -> app2(app2(div, app2(D, x)), x)
app2(D, app2(app2(pow, x), y)) -> app2(app2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))), app2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y)))

The set Q consists of the following terms:

app2(D, t)
app2(D, constant)
app2(D, app2(app2(+, x0), x1))
app2(D, app2(app2(*, x0), x1))
app2(D, app2(app2(-, x0), x1))
app2(D, app2(minus, x0))
app2(D, app2(app2(div, x0), x1))
app2(D, app2(ln, x0))
app2(D, app2(app2(pow, 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:

APP2(D, app2(app2(-, x), y)) -> APP2(D, y)
APP2(D, app2(minus, x)) -> APP2(minus, app2(D, x))
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))
APP2(D, app2(app2(*, x), y)) -> APP2(D, x)
APP2(D, app2(app2(div, x), y)) -> APP2(app2(div, app2(D, x)), y)
APP2(D, app2(app2(pow, x), y)) -> APP2(ln, x)
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(*, app2(app2(pow, x), y)), app2(ln, x))
APP2(D, app2(app2(div, x), y)) -> APP2(app2(*, x), app2(D, y))
APP2(D, app2(app2(+, x), y)) -> APP2(+, app2(D, x))
APP2(D, app2(app2(div, x), y)) -> APP2(pow, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))), app2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y)))
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(pow, x), app2(app2(-, y), 1))
APP2(D, app2(app2(-, x), y)) -> APP2(-, app2(D, x))
APP2(D, app2(app2(pow, x), y)) -> APP2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x)))
APP2(D, app2(app2(pow, x), y)) -> APP2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x)))
APP2(D, app2(app2(*, x), y)) -> APP2(+, app2(app2(*, y), app2(D, x)))
APP2(D, app2(app2(div, x), y)) -> APP2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2))
APP2(D, app2(app2(div, x), y)) -> APP2(div, app2(app2(*, x), app2(D, y)))
APP2(D, app2(minus, x)) -> APP2(D, x)
APP2(D, app2(app2(div, x), y)) -> APP2(-, app2(app2(div, app2(D, x)), y))
APP2(D, app2(app2(div, x), y)) -> APP2(div, app2(D, x))
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))
APP2(D, app2(ln, x)) -> APP2(div, app2(D, x))
APP2(D, app2(ln, x)) -> APP2(app2(div, app2(D, x)), x)
APP2(D, app2(app2(div, x), y)) -> APP2(D, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(D, x)
APP2(D, app2(app2(+, x), y)) -> APP2(D, x)
APP2(D, app2(app2(-, x), y)) -> APP2(app2(-, app2(D, x)), app2(D, y))
APP2(D, app2(app2(div, x), y)) -> APP2(app2(pow, y), 2)
APP2(D, app2(app2(div, x), y)) -> APP2(*, x)
APP2(D, app2(app2(*, x), y)) -> APP2(D, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(D, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y))
APP2(D, app2(ln, x)) -> APP2(D, x)
APP2(D, app2(app2(pow, x), y)) -> APP2(*, app2(app2(pow, x), y))
APP2(D, app2(app2(pow, x), y)) -> APP2(app2(-, y), 1)
APP2(D, app2(app2(*, x), y)) -> APP2(app2(+, app2(app2(*, y), app2(D, x))), app2(app2(*, x), app2(D, y)))
APP2(D, app2(app2(-, x), y)) -> APP2(D, x)
APP2(D, app2(app2(*, x), y)) -> APP2(app2(*, x), app2(D, y))
APP2(D, app2(app2(*, x), y)) -> APP2(app2(*, y), app2(D, x))
APP2(D, app2(app2(pow, x), y)) -> APP2(*, y)
APP2(D, app2(app2(div, x), y)) -> APP2(app2(-, app2(app2(div, app2(D, x)), y)), app2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2)))
APP2(D, app2(app2(*, x), y)) -> APP2(*, y)
APP2(D, app2(app2(div, x), y)) -> APP2(D, x)
APP2(D, app2(app2(pow, x), y)) -> APP2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1))))
APP2(D, app2(app2(pow, x), y)) -> APP2(-, y)
APP2(D, app2(app2(+, x), y)) -> APP2(D, y)
APP2(D, app2(app2(+, x), y)) -> APP2(app2(+, app2(D, x)), app2(D, y))

The TRS R consists of the following rules:

app2(D, t) -> 1
app2(D, constant) -> 0
app2(D, app2(app2(+, x), y)) -> app2(app2(+, app2(D, x)), app2(D, y))
app2(D, app2(app2(*, x), y)) -> app2(app2(+, app2(app2(*, y), app2(D, x))), app2(app2(*, x), app2(D, y)))
app2(D, app2(app2(-, x), y)) -> app2(app2(-, app2(D, x)), app2(D, y))
app2(D, app2(minus, x)) -> app2(minus, app2(D, x))
app2(D, app2(app2(div, x), y)) -> app2(app2(-, app2(app2(div, app2(D, x)), y)), app2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2)))
app2(D, app2(ln, x)) -> app2(app2(div, app2(D, x)), x)
app2(D, app2(app2(pow, x), y)) -> app2(app2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))), app2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y)))

The set Q consists of the following terms:

app2(D, t)
app2(D, constant)
app2(D, app2(app2(+, x0), x1))
app2(D, app2(app2(*, x0), x1))
app2(D, app2(app2(-, x0), x1))
app2(D, app2(minus, x0))
app2(D, app2(app2(div, x0), x1))
app2(D, app2(ln, x0))
app2(D, app2(app2(pow, x0), x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 36 less nodes.

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

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

APP2(D, app2(app2(pow, x), y)) -> APP2(D, y)
APP2(D, app2(ln, x)) -> APP2(D, x)
APP2(D, app2(app2(-, x), y)) -> APP2(D, y)
APP2(D, app2(app2(-, x), y)) -> APP2(D, x)
APP2(D, app2(app2(*, x), y)) -> APP2(D, x)
APP2(D, app2(app2(div, x), y)) -> APP2(D, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(D, x)
APP2(D, app2(app2(div, x), y)) -> APP2(D, x)
APP2(D, app2(app2(+, x), y)) -> APP2(D, x)
APP2(D, app2(app2(+, x), y)) -> APP2(D, y)
APP2(D, app2(minus, x)) -> APP2(D, x)
APP2(D, app2(app2(*, x), y)) -> APP2(D, y)

The TRS R consists of the following rules:

app2(D, t) -> 1
app2(D, constant) -> 0
app2(D, app2(app2(+, x), y)) -> app2(app2(+, app2(D, x)), app2(D, y))
app2(D, app2(app2(*, x), y)) -> app2(app2(+, app2(app2(*, y), app2(D, x))), app2(app2(*, x), app2(D, y)))
app2(D, app2(app2(-, x), y)) -> app2(app2(-, app2(D, x)), app2(D, y))
app2(D, app2(minus, x)) -> app2(minus, app2(D, x))
app2(D, app2(app2(div, x), y)) -> app2(app2(-, app2(app2(div, app2(D, x)), y)), app2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2)))
app2(D, app2(ln, x)) -> app2(app2(div, app2(D, x)), x)
app2(D, app2(app2(pow, x), y)) -> app2(app2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))), app2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y)))

The set Q consists of the following terms:

app2(D, t)
app2(D, constant)
app2(D, app2(app2(+, x0), x1))
app2(D, app2(app2(*, x0), x1))
app2(D, app2(app2(-, x0), x1))
app2(D, app2(minus, x0))
app2(D, app2(app2(div, x0), x1))
app2(D, app2(ln, x0))
app2(D, app2(app2(pow, x0), x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be strictly oriented and are deleted.


APP2(D, app2(app2(pow, x), y)) -> APP2(D, y)
APP2(D, app2(ln, x)) -> APP2(D, x)
APP2(D, app2(app2(-, x), y)) -> APP2(D, y)
APP2(D, app2(app2(-, x), y)) -> APP2(D, x)
APP2(D, app2(app2(*, x), y)) -> APP2(D, x)
APP2(D, app2(app2(div, x), y)) -> APP2(D, y)
APP2(D, app2(app2(pow, x), y)) -> APP2(D, x)
APP2(D, app2(app2(div, x), y)) -> APP2(D, x)
APP2(D, app2(app2(+, x), y)) -> APP2(D, x)
APP2(D, app2(app2(+, x), y)) -> APP2(D, y)
APP2(D, app2(minus, x)) -> APP2(D, x)
APP2(D, app2(app2(*, x), y)) -> APP2(D, y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP2(x1, x2)  =  x2
D  =  D
app2(x1, x2)  =  app2(x1, x2)
pow  =  pow
ln  =  ln
-  =  -
*  =  *
div  =  div
+  =  +
minus  =  minus

Lexicographic Path Order [19].
Precedence:
pow > [D, ln, -, minus]
div > [D, ln, -, minus]
+ > [D, ln, -, minus]


The following usable rules [14] were oriented: none



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

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

app2(D, t) -> 1
app2(D, constant) -> 0
app2(D, app2(app2(+, x), y)) -> app2(app2(+, app2(D, x)), app2(D, y))
app2(D, app2(app2(*, x), y)) -> app2(app2(+, app2(app2(*, y), app2(D, x))), app2(app2(*, x), app2(D, y)))
app2(D, app2(app2(-, x), y)) -> app2(app2(-, app2(D, x)), app2(D, y))
app2(D, app2(minus, x)) -> app2(minus, app2(D, x))
app2(D, app2(app2(div, x), y)) -> app2(app2(-, app2(app2(div, app2(D, x)), y)), app2(app2(div, app2(app2(*, x), app2(D, y))), app2(app2(pow, y), 2)))
app2(D, app2(ln, x)) -> app2(app2(div, app2(D, x)), x)
app2(D, app2(app2(pow, x), y)) -> app2(app2(+, app2(app2(*, app2(app2(*, y), app2(app2(pow, x), app2(app2(-, y), 1)))), app2(D, x))), app2(app2(*, app2(app2(*, app2(app2(pow, x), y)), app2(ln, x))), app2(D, y)))

The set Q consists of the following terms:

app2(D, t)
app2(D, constant)
app2(D, app2(app2(+, x0), x1))
app2(D, app2(app2(*, x0), x1))
app2(D, app2(app2(-, x0), x1))
app2(D, app2(minus, x0))
app2(D, app2(app2(div, x0), x1))
app2(D, app2(ln, x0))
app2(D, app2(app2(pow, 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.