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:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact


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

APP(app(add, app(s, x)), y) → APP(app(add, x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(mult, app(s, x)), y) → APP(add, app(app(mult, x), y))
APP(app(mult, app(s, x)), y) → APP(app(add, app(app(mult, x), y)), y)
APP(app(add, app(s, x)), y) → APP(s, app(app(add, x), y))
APP(app(add, app(s, x)), y) → APP(add, x)
FACT → APP(s, 0)
FACT → APP(rec, mult)
APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
FACT → APP(app(rec, mult), app(s, 0))
APP(app(mult, app(s, x)), y) → APP(mult, x)
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof

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

APP(app(add, app(s, x)), y) → APP(app(add, x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(mult, app(s, x)), y) → APP(add, app(app(mult, x), y))
APP(app(mult, app(s, x)), y) → APP(app(add, app(app(mult, x), y)), y)
APP(app(add, app(s, x)), y) → APP(s, app(app(add, x), y))
APP(app(add, app(s, x)), y) → APP(add, x)
FACT → APP(s, 0)
FACT → APP(rec, mult)
APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
FACT → APP(app(rec, mult), app(s, 0))
APP(app(mult, app(s, x)), y) → APP(mult, x)
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof

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

APP(app(add, app(s, x)), y) → APP(app(add, x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(mult, app(s, x)), y) → APP(add, app(app(mult, x), y))
APP(app(mult, app(s, x)), y) → APP(app(add, app(app(mult, x), y)), y)
APP(app(add, app(s, x)), y) → APP(s, app(app(add, x), y))
APP(app(add, app(s, x)), y) → APP(add, x)
FACT → APP(s, 0)
FACT → APP(rec, mult)
FACT → APP(app(rec, mult), app(s, 0))
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)
APP(app(mult, app(s, x)), y) → APP(mult, x)
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 8 less nodes.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
                  ↳ QDP
                  ↳ QDP

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

APP(app(add, app(s, x)), y) → APP(app(add, x), y)

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13]. Here, we combined the reduction pair processor with the A-transformation [14] which results in the following intermediate Q-DP Problem.
Q DP problem:
The TRS P consists of the following rules:

ADD(s(x), y) → ADD(x, y)

R is empty.
The set Q consists of the following terms:

add(0, x0)
add(s(x0), x1)
mult(0, x0)
mult(s(x0), x1)
rec(x0, x1, 0)
rec(x0, x1, s(x2))
fact

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


The following pairs can be oriented strictly and are deleted.


APP(app(add, app(s, x)), y) → APP(app(add, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
ADD(x1, x2)  =  x1
s(x1)  =  s(x1)

Recursive Path Order [2].
Precedence:
trivial

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
                  ↳ QDP
                  ↳ QDP

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

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
                  ↳ QDP

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

APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13]. Here, we combined the reduction pair processor with the A-transformation [14] which results in the following intermediate Q-DP Problem.
Q DP problem:
The TRS P consists of the following rules:

MULT(s(x), y) → MULT(x, y)

R is empty.
The set Q consists of the following terms:

add(0, x0)
add(s(x0), x1)
mult(0, x0)
mult(s(x0), x1)
rec(x0, x1, 0)
rec(x0, x1, s(x2))
fact

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


The following pairs can be oriented strictly and are deleted.


APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
MULT(x1, x2)  =  x1
s(x1)  =  s(x1)

Recursive Path Order [2].
Precedence:
trivial

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
                  ↳ QDP

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

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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