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:

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

Q is empty.


QTRS
  ↳ Non-Overlap Check

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

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

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:

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

The set Q consists of the following terms:

eq2(0, 0)
eq2(0, s1(x0))
eq2(s1(x0), 0)
eq2(s1(x0), s1(x1))
or2(true, x0)
or2(false, x0)
union2(empty, x0)
union2(edge3(x0, x1, x2), x3)
reach4(x0, x1, empty, x2)
reach4(x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(false, x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(false, x0, x1, edge3(x2, x3, x4), x5)


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

REACH4(x, y, edge3(u, v, i), h) -> IF_REACH_15(eq2(x, u), x, y, edge3(u, v, i), h)
IF_REACH_15(false, x, y, edge3(u, v, i), h) -> REACH4(x, y, i, edge3(u, v, h))
UNION2(edge3(x, y, i), h) -> UNION2(i, h)
EQ2(s1(x), s1(y)) -> EQ2(x, y)
IF_REACH_15(true, x, y, edge3(u, v, i), h) -> EQ2(y, v)
IF_REACH_15(true, x, y, edge3(u, v, i), h) -> IF_REACH_25(eq2(y, v), x, y, edge3(u, v, i), h)
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> REACH4(x, y, i, h)
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> UNION2(i, h)
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> OR2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> REACH4(v, y, union2(i, h), empty)
REACH4(x, y, edge3(u, v, i), h) -> EQ2(x, u)

The TRS R consists of the following rules:

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

The set Q consists of the following terms:

eq2(0, 0)
eq2(0, s1(x0))
eq2(s1(x0), 0)
eq2(s1(x0), s1(x1))
or2(true, x0)
or2(false, x0)
union2(empty, x0)
union2(edge3(x0, x1, x2), x3)
reach4(x0, x1, empty, x2)
reach4(x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(false, x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(false, x0, x1, edge3(x2, x3, x4), x5)

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:

REACH4(x, y, edge3(u, v, i), h) -> IF_REACH_15(eq2(x, u), x, y, edge3(u, v, i), h)
IF_REACH_15(false, x, y, edge3(u, v, i), h) -> REACH4(x, y, i, edge3(u, v, h))
UNION2(edge3(x, y, i), h) -> UNION2(i, h)
EQ2(s1(x), s1(y)) -> EQ2(x, y)
IF_REACH_15(true, x, y, edge3(u, v, i), h) -> EQ2(y, v)
IF_REACH_15(true, x, y, edge3(u, v, i), h) -> IF_REACH_25(eq2(y, v), x, y, edge3(u, v, i), h)
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> REACH4(x, y, i, h)
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> UNION2(i, h)
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> OR2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> REACH4(v, y, union2(i, h), empty)
REACH4(x, y, edge3(u, v, i), h) -> EQ2(x, u)

The TRS R consists of the following rules:

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

The set Q consists of the following terms:

eq2(0, 0)
eq2(0, s1(x0))
eq2(s1(x0), 0)
eq2(s1(x0), s1(x1))
or2(true, x0)
or2(false, x0)
union2(empty, x0)
union2(edge3(x0, x1, x2), x3)
reach4(x0, x1, empty, x2)
reach4(x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(false, x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(false, x0, x1, edge3(x2, x3, x4), x5)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 4 less nodes.

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

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

UNION2(edge3(x, y, i), h) -> UNION2(i, h)

The TRS R consists of the following rules:

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

The set Q consists of the following terms:

eq2(0, 0)
eq2(0, s1(x0))
eq2(s1(x0), 0)
eq2(s1(x0), s1(x1))
or2(true, x0)
or2(false, x0)
union2(empty, x0)
union2(edge3(x0, x1, x2), x3)
reach4(x0, x1, empty, x2)
reach4(x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(false, x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(false, x0, x1, edge3(x2, x3, x4), x5)

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.


UNION2(edge3(x, y, i), h) -> UNION2(i, h)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
UNION2(x1, x2)  =  UNION1(x1)
edge3(x1, x2, x3)  =  edge1(x3)

Lexicographic Path Order [19].
Precedence:
edge1 > UNION1


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Non-Overlap Check
    ↳ 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:

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

The set Q consists of the following terms:

eq2(0, 0)
eq2(0, s1(x0))
eq2(s1(x0), 0)
eq2(s1(x0), s1(x1))
or2(true, x0)
or2(false, x0)
union2(empty, x0)
union2(edge3(x0, x1, x2), x3)
reach4(x0, x1, empty, x2)
reach4(x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(false, x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(false, x0, x1, edge3(x2, x3, x4), x5)

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
                ↳ QDPOrderProof
              ↳ QDP

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

EQ2(s1(x), s1(y)) -> EQ2(x, y)

The TRS R consists of the following rules:

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

The set Q consists of the following terms:

eq2(0, 0)
eq2(0, s1(x0))
eq2(s1(x0), 0)
eq2(s1(x0), s1(x1))
or2(true, x0)
or2(false, x0)
union2(empty, x0)
union2(edge3(x0, x1, x2), x3)
reach4(x0, x1, empty, x2)
reach4(x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(false, x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(false, x0, x1, edge3(x2, x3, x4), x5)

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.


EQ2(s1(x), s1(y)) -> EQ2(x, y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
EQ2(x1, x2)  =  EQ1(x1)
s1(x1)  =  s1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Non-Overlap Check
    ↳ 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:

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

The set Q consists of the following terms:

eq2(0, 0)
eq2(0, s1(x0))
eq2(s1(x0), 0)
eq2(s1(x0), s1(x1))
or2(true, x0)
or2(false, x0)
union2(empty, x0)
union2(edge3(x0, x1, x2), x3)
reach4(x0, x1, empty, x2)
reach4(x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(false, x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(false, x0, x1, edge3(x2, x3, x4), x5)

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
QDP

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

REACH4(x, y, edge3(u, v, i), h) -> IF_REACH_15(eq2(x, u), x, y, edge3(u, v, i), h)
IF_REACH_15(false, x, y, edge3(u, v, i), h) -> REACH4(x, y, i, edge3(u, v, h))
IF_REACH_15(true, x, y, edge3(u, v, i), h) -> IF_REACH_25(eq2(y, v), x, y, edge3(u, v, i), h)
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> REACH4(x, y, i, h)
IF_REACH_25(false, x, y, edge3(u, v, i), h) -> REACH4(v, y, union2(i, h), empty)

The TRS R consists of the following rules:

eq2(0, 0) -> true
eq2(0, s1(x)) -> false
eq2(s1(x), 0) -> false
eq2(s1(x), s1(y)) -> eq2(x, y)
or2(true, y) -> true
or2(false, y) -> y
union2(empty, h) -> h
union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
reach4(x, y, empty, h) -> false
reach4(x, y, edge3(u, v, i), h) -> if_reach_15(eq2(x, u), x, y, edge3(u, v, i), h)
if_reach_15(true, x, y, edge3(u, v, i), h) -> if_reach_25(eq2(y, v), x, y, edge3(u, v, i), h)
if_reach_25(true, x, y, edge3(u, v, i), h) -> true
if_reach_25(false, x, y, edge3(u, v, i), h) -> or2(reach4(x, y, i, h), reach4(v, y, union2(i, h), empty))
if_reach_15(false, x, y, edge3(u, v, i), h) -> reach4(x, y, i, edge3(u, v, h))

The set Q consists of the following terms:

eq2(0, 0)
eq2(0, s1(x0))
eq2(s1(x0), 0)
eq2(s1(x0), s1(x1))
or2(true, x0)
or2(false, x0)
union2(empty, x0)
union2(edge3(x0, x1, x2), x3)
reach4(x0, x1, empty, x2)
reach4(x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(true, x0, x1, edge3(x2, x3, x4), x5)
if_reach_25(false, x0, x1, edge3(x2, x3, x4), x5)
if_reach_15(false, x0, x1, edge3(x2, x3, x4), x5)

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