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:

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

Q is empty.


QTRS
  ↳ Overlay + Local Confluence

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

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

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:

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
if(true, x0, x1)
if(false, x0, x1)
add(0, x0)
add(s(x0), x1)
first(0, x0)
first(s(x0), cons(x1, x2))
from(x0)


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

ADD(s(X), Y) → ADD(X, Y)
FIRST(s(X), cons(Y, Z)) → FIRST(X, Z)
FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
if(true, x0, x1)
if(false, x0, x1)
add(0, x0)
add(s(x0), x1)
first(0, x0)
first(s(x0), cons(x1, x2))
from(x0)

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:

ADD(s(X), Y) → ADD(X, Y)
FIRST(s(X), cons(Y, Z)) → FIRST(X, Z)
FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
if(true, x0, x1)
if(false, x0, x1)
add(0, x0)
add(s(x0), x1)
first(0, x0)
first(s(x0), cons(x1, x2))
from(x0)

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:

ADD(s(X), Y) → ADD(X, Y)
FIRST(s(X), cons(Y, Z)) → FIRST(X, Z)
FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
if(true, x0, x1)
if(false, x0, x1)
add(0, x0)
add(s(x0), x1)
first(0, x0)
first(s(x0), cons(x1, x2))
from(x0)

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

↳ 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:

FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
if(true, x0, x1)
if(false, x0, x1)
add(0, x0)
add(s(x0), x1)
first(0, x0)
first(s(x0), cons(x1, x2))
from(x0)

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

↳ 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:

FIRST(s(X), cons(Y, Z)) → FIRST(X, Z)

The TRS R consists of the following rules:

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
if(true, x0, x1)
if(false, x0, x1)
add(0, x0)
add(s(x0), x1)
first(0, x0)
first(s(x0), cons(x1, x2))
from(x0)

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.


FIRST(s(X), cons(Y, Z)) → FIRST(X, Z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
FIRST(x1, x2)  =  x2
cons(x1, x2)  =  cons(x2)

Lexicographic path order with status [19].
Precedence:
trivial

Status:
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:

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
if(true, x0, x1)
if(false, x0, x1)
add(0, x0)
add(s(x0), x1)
first(0, x0)
first(s(x0), cons(x1, x2))
from(x0)

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
                    ↳ QDPOrderProof

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

ADD(s(X), Y) → ADD(X, Y)

The TRS R consists of the following rules:

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
if(true, x0, x1)
if(false, x0, x1)
add(0, x0)
add(s(x0), x1)
first(0, x0)
first(s(x0), cons(x1, x2))
from(x0)

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.


ADD(s(X), Y) → 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)

Lexicographic path order with status [19].
Precedence:
trivial

Status:
trivial

The following usable rules [14] were oriented: none



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

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

and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
if(true, x0, x1)
if(false, x0, x1)
add(0, x0)
add(s(x0), x1)
first(0, x0)
first(s(x0), cons(x1, x2))
from(x0)

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