Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
    ↳5 QDP
        ↳6 QDPOrderProof (⇔)
        ↳7 QDP
        ↳8 PisEmptyProof (⇔)
        ↳9 TRUE
    ↳10 QDP
        ↳11 QDPOrderProof (⇔)
        ↳12 QDP
        ↳13 PisEmptyProof (⇔)
        ↳14 TRUE
    ↳15 QDP
        ↳16 QDPOrderProof (⇔)
        ↳17 QDP
        ↳18 DependencyGraphProof (⇔)
        ↳19 QDP
        ↳20 QDPOrderProof (⇔)
        ↳21 QDP
        ↳22 DependencyGraphProof (⇔)
        ↳23 TRUE

(0) Obligation:

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

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

EQ(s(x), s(y)) → EQ(x, y)
UNION(edge(x, y, i), h) → UNION(i, h)
REACH(x, y, edge(u, v, i), h) → IF_REACH_1(eq(x, u), x, y, edge(u, v, i), h)
REACH(x, y, edge(u, v, i), h) → EQ(x, u)
IF_REACH_1(true, x, y, edge(u, v, i), h) → IF_REACH_2(eq(y, v), x, y, edge(u, v, i), h)
IF_REACH_1(true, x, y, edge(u, v, i), h) → EQ(y, v)
IF_REACH_2(false, x, y, edge(u, v, i), h) → OR(reach(x, y, i, h), reach(v, y, union(i, h), empty))
IF_REACH_2(false, x, y, edge(u, v, i), h) → REACH(x, y, i, h)
IF_REACH_2(false, x, y, edge(u, v, i), h) → REACH(v, y, union(i, h), empty)
IF_REACH_2(false, x, y, edge(u, v, i), h) → UNION(i, h)
IF_REACH_1(false, x, y, edge(u, v, i), h) → REACH(x, y, i, edge(u, v, h))

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 4 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

UNION(edge(x, y, i), h) → UNION(i, h)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


UNION(edge(x, y, i), h) → UNION(i, h)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(UNION(x1, x2)) = x1   
POL(edge(x1, x2, x3)) = 1 + x3   

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

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

EQ(s(x), s(y)) → EQ(x, y)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


EQ(s(x), s(y)) → EQ(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(EQ(x1, x2)) = x2   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(14) TRUE

(15) Obligation:

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

REACH(x, y, edge(u, v, i), h) → IF_REACH_1(eq(x, u), x, y, edge(u, v, i), h)
IF_REACH_1(true, x, y, edge(u, v, i), h) → IF_REACH_2(eq(y, v), x, y, edge(u, v, i), h)
IF_REACH_2(false, x, y, edge(u, v, i), h) → REACH(x, y, i, h)
IF_REACH_2(false, x, y, edge(u, v, i), h) → REACH(v, y, union(i, h), empty)
IF_REACH_1(false, x, y, edge(u, v, i), h) → REACH(x, y, i, edge(u, v, h))

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


IF_REACH_2(false, x, y, edge(u, v, i), h) → REACH(x, y, i, h)
IF_REACH_2(false, x, y, edge(u, v, i), h) → REACH(v, y, union(i, h), empty)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(IF_REACH_1(x1, x2, x3, x4, x5)) = x4 + x5   
POL(IF_REACH_2(x1, x2, x3, x4, x5)) = x4 + x5   
POL(REACH(x1, x2, x3, x4)) = x3 + x4   
POL(edge(x1, x2, x3)) = 1 + x3   
POL(empty) = 0   
POL(eq(x1, x2)) = 0   
POL(false) = 0   
POL(s(x1)) = 0   
POL(true) = 0   
POL(union(x1, x2)) = x1 + x2   

The following usable rules [FROCOS05] were oriented:

union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))

(17) Obligation:

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

REACH(x, y, edge(u, v, i), h) → IF_REACH_1(eq(x, u), x, y, edge(u, v, i), h)
IF_REACH_1(true, x, y, edge(u, v, i), h) → IF_REACH_2(eq(y, v), x, y, edge(u, v, i), h)
IF_REACH_1(false, x, y, edge(u, v, i), h) → REACH(x, y, i, edge(u, v, h))

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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

(18) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(19) Obligation:

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

IF_REACH_1(false, x, y, edge(u, v, i), h) → REACH(x, y, i, edge(u, v, h))
REACH(x, y, edge(u, v, i), h) → IF_REACH_1(eq(x, u), x, y, edge(u, v, i), h)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


REACH(x, y, edge(u, v, i), h) → IF_REACH_1(eq(x, u), x, y, edge(u, v, i), h)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(IF_REACH_1(x1, x2, x3, x4, x5)) = x2 + x4   
POL(REACH(x1, x2, x3, x4)) = 1 + x1 + x3   
POL(edge(x1, x2, x3)) = 1 + x3   
POL(eq(x1, x2)) = 0   
POL(false) = 0   
POL(s(x1)) = 0   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

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

IF_REACH_1(false, x, y, edge(u, v, i), h) → REACH(x, y, i, edge(u, v, h))

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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

(22) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(23) TRUE