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 Order [17,21] with Interpretation:

POL( UNION2(x1, x2) ) = x1 + x2


POL( edge3(x1, ..., x3) ) = x3 + 1



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 Order [17,21] with Interpretation:

POL( s1(x1) ) = x1 + 1


POL( EQ2(x1, x2) ) = 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 Order [17,21] with Interpretation:

POL( true ) = max{0, -1}


POL( edge3(x1, ..., x3) ) = x3 + 1


POL( IF_REACH_25(x1, ..., x5) ) = x4 + x5


POL( union2(x1, x2) ) = x1 + x2


POL( false ) = 1


POL( REACH4(x1, ..., x4) ) = x3 + x4


POL( eq2(x1, x2) ) = 0


POL( s1(x1) ) = x1 + 1


POL( 0 ) = 1


POL( empty ) = 0


POL( IF_REACH_15(x1, ..., x5) ) = x4 + x5



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.


IF_REACH_15(false, x, y, edge3(u, v, i), h) -> REACH4(x, y, i, edge3(u, v, h))
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)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( true ) = max{0, -1}


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


POL( false ) = 1


POL( REACH4(x1, ..., x4) ) = x2 + x3 + 1


POL( eq2(x1, x2) ) = max{0, x2 - 1}


POL( 0 ) = max{0, -1}


POL( s1(x1) ) = max{0, x1 - 1}


POL( IF_REACH_15(x1, ..., x5) ) = x3 + x4 + 1



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:

REACH4(x, y, edge3(u, v, i), h) -> IF_REACH_15(eq2(x, u), 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 0 SCCs with 1 less node.