(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))
isEmpty(empty) → true
isEmpty(edge(x, y, i)) → false
from(edge(x, y, i)) → x
to(edge(x, y, i)) → y
rest(edge(x, y, i)) → i
rest(empty) → empty
reach(x, y, i, h) → if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
if1(true, b1, b2, b3, x, y, i, h) → true
if1(false, b1, b2, b3, x, y, i, h) → if2(b1, b2, b3, x, y, i, h)
if2(true, b2, b3, x, y, i, h) → false
if2(false, b2, b3, x, y, i, h) → if3(b2, b3, x, y, i, h)
if3(false, b3, x, y, i, h) → reach(x, y, rest(i), edge(from(i), to(i), h))
if3(true, b3, x, y, i, h) → if4(b3, x, y, i, h)
if4(true, x, y, i, h) → true
if4(false, x, y, i, h) → or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))
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))
isEmpty(empty) → true
isEmpty(edge(x, y, i)) → false
from(edge(x, y, i)) → x
to(edge(x, y, i)) → y
rest(edge(x, y, i)) → i
rest(empty) → empty
reach(x, y, i, h) → if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
if1(true, b1, b2, b3, x, y, i, h) → true
if1(false, b1, b2, b3, x, y, i, h) → if2(b1, b2, b3, x, y, i, h)
if2(true, b2, b3, x, y, i, h) → false
if2(false, b2, b3, x, y, i, h) → if3(b2, b3, x, y, i, h)
if3(false, b3, x, y, i, h) → reach(x, y, rest(i), edge(from(i), to(i), h))
if3(true, b3, x, y, i, h) → if4(b3, x, y, i, h)
if4(true, x, y, i, h) → true
if4(false, x, y, i, h) → or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))
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)
isEmpty(empty)
isEmpty(edge(x0, x1, x2))
from(edge(x0, x1, x2))
to(edge(x0, x1, x2))
rest(edge(x0, x1, x2))
rest(empty)
reach(x0, x1, x2, x3)
if1(true, x0, x1, x2, x3, x4, x5, x6)
if1(false, x0, x1, x2, x3, x4, x5, x6)
if2(true, x0, x1, x2, x3, x4, x5)
if2(false, x0, x1, x2, x3, x4, x5)
if3(false, x0, x1, x2, x3, x4)
if3(true, x0, x1, x2, x3, x4)
if4(true, x0, x1, x2, x3)
if4(false, x0, x1, x2, x3)
(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, i, h) → IF1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
REACH(x, y, i, h) → EQ(x, y)
REACH(x, y, i, h) → ISEMPTY(i)
REACH(x, y, i, h) → EQ(x, from(i))
REACH(x, y, i, h) → FROM(i)
REACH(x, y, i, h) → EQ(y, to(i))
REACH(x, y, i, h) → TO(i)
IF1(false, b1, b2, b3, x, y, i, h) → IF2(b1, b2, b3, x, y, i, h)
IF2(false, b2, b3, x, y, i, h) → IF3(b2, b3, x, y, i, h)
IF3(false, b3, x, y, i, h) → REACH(x, y, rest(i), edge(from(i), to(i), h))
IF3(false, b3, x, y, i, h) → REST(i)
IF3(false, b3, x, y, i, h) → FROM(i)
IF3(false, b3, x, y, i, h) → TO(i)
IF3(true, b3, x, y, i, h) → IF4(b3, x, y, i, h)
IF4(false, x, y, i, h) → OR(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))
IF4(false, x, y, i, h) → REACH(x, y, rest(i), h)
IF4(false, x, y, i, h) → REST(i)
IF4(false, x, y, i, h) → REACH(to(i), y, union(rest(i), h), empty)
IF4(false, x, y, i, h) → TO(i)
IF4(false, x, y, i, h) → UNION(rest(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))
isEmpty(empty) → true
isEmpty(edge(x, y, i)) → false
from(edge(x, y, i)) → x
to(edge(x, y, i)) → y
rest(edge(x, y, i)) → i
rest(empty) → empty
reach(x, y, i, h) → if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
if1(true, b1, b2, b3, x, y, i, h) → true
if1(false, b1, b2, b3, x, y, i, h) → if2(b1, b2, b3, x, y, i, h)
if2(true, b2, b3, x, y, i, h) → false
if2(false, b2, b3, x, y, i, h) → if3(b2, b3, x, y, i, h)
if3(false, b3, x, y, i, h) → reach(x, y, rest(i), edge(from(i), to(i), h))
if3(true, b3, x, y, i, h) → if4(b3, x, y, i, h)
if4(true, x, y, i, h) → true
if4(false, x, y, i, h) → or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))
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)
isEmpty(empty)
isEmpty(edge(x0, x1, x2))
from(edge(x0, x1, x2))
to(edge(x0, x1, x2))
rest(edge(x0, x1, x2))
rest(empty)
reach(x0, x1, x2, x3)
if1(true, x0, x1, x2, x3, x4, x5, x6)
if1(false, x0, x1, x2, x3, x4, x5, x6)
if2(true, x0, x1, x2, x3, x4, x5)
if2(false, x0, x1, x2, x3, x4, x5)
if3(false, x0, x1, x2, x3, x4)
if3(true, x0, x1, x2, x3, x4)
if4(true, x0, x1, x2, x3)
if4(false, x0, x1, x2, x3)
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 13 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))
isEmpty(empty) → true
isEmpty(edge(x, y, i)) → false
from(edge(x, y, i)) → x
to(edge(x, y, i)) → y
rest(edge(x, y, i)) → i
rest(empty) → empty
reach(x, y, i, h) → if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
if1(true, b1, b2, b3, x, y, i, h) → true
if1(false, b1, b2, b3, x, y, i, h) → if2(b1, b2, b3, x, y, i, h)
if2(true, b2, b3, x, y, i, h) → false
if2(false, b2, b3, x, y, i, h) → if3(b2, b3, x, y, i, h)
if3(false, b3, x, y, i, h) → reach(x, y, rest(i), edge(from(i), to(i), h))
if3(true, b3, x, y, i, h) → if4(b3, x, y, i, h)
if4(true, x, y, i, h) → true
if4(false, x, y, i, h) → or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))
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)
isEmpty(empty)
isEmpty(edge(x0, x1, x2))
from(edge(x0, x1, x2))
to(edge(x0, x1, x2))
rest(edge(x0, x1, x2))
rest(empty)
reach(x0, x1, x2, x3)
if1(true, x0, x1, x2, x3, x4, x5, x6)
if1(false, x0, x1, x2, x3, x4, x5, x6)
if2(true, x0, x1, x2, x3, x4, x5)
if2(false, x0, x1, x2, x3, x4, x5)
if3(false, x0, x1, x2, x3, x4)
if3(true, x0, x1, x2, x3, x4)
if4(true, x0, x1, x2, x3)
if4(false, x0, x1, x2, x3)
We have to consider all minimal (P,Q,R)-chains.
(8) 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))
isEmpty(empty) → true
isEmpty(edge(x, y, i)) → false
from(edge(x, y, i)) → x
to(edge(x, y, i)) → y
rest(edge(x, y, i)) → i
rest(empty) → empty
reach(x, y, i, h) → if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
if1(true, b1, b2, b3, x, y, i, h) → true
if1(false, b1, b2, b3, x, y, i, h) → if2(b1, b2, b3, x, y, i, h)
if2(true, b2, b3, x, y, i, h) → false
if2(false, b2, b3, x, y, i, h) → if3(b2, b3, x, y, i, h)
if3(false, b3, x, y, i, h) → reach(x, y, rest(i), edge(from(i), to(i), h))
if3(true, b3, x, y, i, h) → if4(b3, x, y, i, h)
if4(true, x, y, i, h) → true
if4(false, x, y, i, h) → or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))
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)
isEmpty(empty)
isEmpty(edge(x0, x1, x2))
from(edge(x0, x1, x2))
to(edge(x0, x1, x2))
rest(edge(x0, x1, x2))
rest(empty)
reach(x0, x1, x2, x3)
if1(true, x0, x1, x2, x3, x4, x5, x6)
if1(false, x0, x1, x2, x3, x4, x5, x6)
if2(true, x0, x1, x2, x3, x4, x5)
if2(false, x0, x1, x2, x3, x4, x5)
if3(false, x0, x1, x2, x3, x4)
if3(true, x0, x1, x2, x3, x4)
if4(true, x0, x1, x2, x3)
if4(false, x0, x1, x2, x3)
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
REACH(x, y, i, h) → IF1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
IF1(false, b1, b2, b3, x, y, i, h) → IF2(b1, b2, b3, x, y, i, h)
IF2(false, b2, b3, x, y, i, h) → IF3(b2, b3, x, y, i, h)
IF3(false, b3, x, y, i, h) → REACH(x, y, rest(i), edge(from(i), to(i), h))
IF3(true, b3, x, y, i, h) → IF4(b3, x, y, i, h)
IF4(false, x, y, i, h) → REACH(x, y, rest(i), h)
IF4(false, x, y, i, h) → REACH(to(i), y, union(rest(i), h), 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))
isEmpty(empty) → true
isEmpty(edge(x, y, i)) → false
from(edge(x, y, i)) → x
to(edge(x, y, i)) → y
rest(edge(x, y, i)) → i
rest(empty) → empty
reach(x, y, i, h) → if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
if1(true, b1, b2, b3, x, y, i, h) → true
if1(false, b1, b2, b3, x, y, i, h) → if2(b1, b2, b3, x, y, i, h)
if2(true, b2, b3, x, y, i, h) → false
if2(false, b2, b3, x, y, i, h) → if3(b2, b3, x, y, i, h)
if3(false, b3, x, y, i, h) → reach(x, y, rest(i), edge(from(i), to(i), h))
if3(true, b3, x, y, i, h) → if4(b3, x, y, i, h)
if4(true, x, y, i, h) → true
if4(false, x, y, i, h) → or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))
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)
isEmpty(empty)
isEmpty(edge(x0, x1, x2))
from(edge(x0, x1, x2))
to(edge(x0, x1, x2))
rest(edge(x0, x1, x2))
rest(empty)
reach(x0, x1, x2, x3)
if1(true, x0, x1, x2, x3, x4, x5, x6)
if1(false, x0, x1, x2, x3, x4, x5, x6)
if2(true, x0, x1, x2, x3, x4, x5)
if2(false, x0, x1, x2, x3, x4, x5)
if3(false, x0, x1, x2, x3, x4)
if3(true, x0, x1, x2, x3, x4)
if4(true, x0, x1, x2, x3)
if4(false, x0, x1, x2, x3)
We have to consider all minimal (P,Q,R)-chains.