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:

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
  ↳ 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))

Q is empty.

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))

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

↳ 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))

Q is empty.
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
  ↳ 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))

Q is empty.
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.


UNION2(edge3(x, y, i), h) -> UNION2(i, h)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(UNION2(x1, x2)) = x1   
POL(edge3(x1, x2, x3)) = 1 + x3   

The following usable rules [14] were oriented: none



↳ 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))

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

↳ 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))

Q is empty.
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.


EQ2(s1(x), s1(y)) -> EQ2(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(EQ2(x1, x2)) = x2   
POL(s1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ 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))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof

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))

Q is empty.
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.


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 remaining pairs can at least be oriented weakly.

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)
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(IF_REACH_15(x1, x2, x3, x4, x5)) = x3 + x4 + x5   
POL(IF_REACH_25(x1, x2, x3, x4, x5)) = x3 + x4 + x5   
POL(REACH4(x1, x2, x3, x4)) = x2 + x3 + x4   
POL(edge3(x1, x2, x3)) = 1 + x3   
POL(empty) = 0   
POL(eq2(x1, x2)) = 0   
POL(false) = 0   
POL(s1(x1)) = 0   
POL(true) = 0   
POL(union2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented:

union2(edge3(x, y, i), h) -> edge3(x, y, union2(i, h))
union2(empty, h) -> h



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
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))
IF_REACH_15(true, x, y, edge3(u, v, i), h) -> IF_REACH_25(eq2(y, v), x, y, edge3(u, v, 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))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
QDP
                    ↳ QDPOrderProof

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))

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.
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.


REACH4(x, y, edge3(u, v, i), h) -> IF_REACH_15(eq2(x, u), x, y, edge3(u, v, i), h)
The remaining pairs can at least be oriented weakly.

IF_REACH_15(false, x, y, edge3(u, v, i), h) -> REACH4(x, y, i, edge3(u, v, h))
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(IF_REACH_15(x1, x2, x3, x4, x5)) = x4   
POL(REACH4(x1, x2, x3, x4)) = 1 + x3   
POL(edge3(x1, x2, x3)) = 1 + x3   
POL(eq2(x1, x2)) = 0   
POL(false) = 0   
POL(s1(x1)) = 0   
POL(true) = 0   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ DependencyGraphProof

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

IF_REACH_15(false, x, y, edge3(u, v, i), h) -> REACH4(x, y, i, edge3(u, v, 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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.