(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) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) 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))

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
union(empty, x0)
union(edge(x0, x1, x2), x3)
reach(x0, x1, empty, x2)
reach(x0, x1, edge(x2, x3, x4), x5)
if_reach_1(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(false, x0, x1, edge(x2, x3, x4), x5)
if_reach_1(false, x0, x1, edge(x2, x3, x4), x5)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

(4) 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))

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
union(empty, x0)
union(edge(x0, x1, x2), x3)
reach(x0, x1, empty, x2)
reach(x0, x1, edge(x2, x3, x4), x5)
if_reach_1(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(false, x0, x1, edge(x2, x3, x4), x5)
if_reach_1(false, x0, x1, edge(x2, x3, x4), x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Complex Obligation (AND)

(7) 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))

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
union(empty, x0)
union(edge(x0, x1, x2), x3)
reach(x0, x1, empty, x2)
reach(x0, x1, edge(x2, x3, x4), x5)
if_reach_1(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(false, x0, x1, edge(x2, x3, x4), x5)
if_reach_1(false, x0, x1, edge(x2, x3, x4), x5)

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

(8) 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: Combined order from the following AFS and order.
UNION(x1, x2)  =  UNION(x1, x2)
edge(x1, x2, x3)  =  edge(x2, x3)
eq(x1, x2)  =  eq(x1, x2)
0  =  0
true  =  true
s(x1)  =  x1
false  =  false
or(x1, x2)  =  x2
union(x1, x2)  =  union(x1, x2)
empty  =  empty
reach(x1, x2, x3, x4)  =  x2
if_reach_1(x1, x2, x3, x4, x5)  =  x3
if_reach_2(x1, x2, x3, x4, x5)  =  x3

Lexicographic Path Order [LPO].
Precedence:
UNION2 > [edge2, true, false, empty]
eq2 > [edge2, true, false, empty]
0 > [edge2, true, false, empty]
union2 > [edge2, true, false, empty]


The following usable rules [FROCOS05] were oriented:

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))

(9) 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))

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
union(empty, x0)
union(edge(x0, x1, x2), x3)
reach(x0, x1, empty, x2)
reach(x0, x1, edge(x2, x3, x4), x5)
if_reach_1(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(false, x0, x1, edge(x2, x3, x4), x5)
if_reach_1(false, x0, x1, edge(x2, x3, x4), x5)

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) 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))

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
union(empty, x0)
union(edge(x0, x1, x2), x3)
reach(x0, x1, empty, x2)
reach(x0, x1, edge(x2, x3, x4), x5)
if_reach_1(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(false, x0, x1, edge(x2, x3, x4), x5)
if_reach_1(false, x0, x1, edge(x2, x3, x4), x5)

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

(13) 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: Combined order from the following AFS and order.
EQ(x1, x2)  =  x2
s(x1)  =  s(x1)
eq(x1, x2)  =  eq(x2)
0  =  0
true  =  true
false  =  false
or(x1, x2)  =  x2
union(x1, x2)  =  union(x1, x2)
empty  =  empty
edge(x1, x2, x3)  =  edge
reach(x1, x2, x3, x4)  =  reach(x2, x4)
if_reach_1(x1, x2, x3, x4, x5)  =  if_reach_1(x3, x5)
if_reach_2(x1, x2, x3, x4, x5)  =  if_reach_2(x3, x4)

Lexicographic Path Order [LPO].
Precedence:
s1 > false > [0, true, empty, edge]
eq1 > false > [0, true, empty, edge]
union2 > [0, true, empty, edge]
[reach2, ifreach12, ifreach22] > false > [0, true, empty, edge]


The following usable rules [FROCOS05] were oriented:

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))

(14) 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))

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
union(empty, x0)
union(edge(x0, x1, x2), x3)
reach(x0, x1, empty, x2)
reach(x0, x1, edge(x2, x3, x4), x5)
if_reach_1(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(false, x0, x1, edge(x2, x3, x4), x5)
if_reach_1(false, x0, x1, edge(x2, x3, x4), x5)

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

(15) PisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE

(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_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))

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
union(empty, x0)
union(edge(x0, x1, x2), x3)
reach(x0, x1, empty, x2)
reach(x0, x1, edge(x2, x3, x4), x5)
if_reach_1(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(false, x0, x1, edge(x2, x3, x4), x5)
if_reach_1(false, x0, x1, edge(x2, x3, x4), x5)

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