(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) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
UNION(edge(x, y, i), h) → UNION(i, h)
R is 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)
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) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
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)
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
UNION(edge(x, y, i), h) → UNION(i, h)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- UNION(edge(x, y, i), h) → UNION(i, h)
The graph contains the following edges 1 > 1, 2 >= 2
(13) YES
(14) 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.
(15) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
EQ(s(x), s(y)) → EQ(x, y)
R is 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)
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.
(17) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
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)
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
EQ(s(x), s(y)) → EQ(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- EQ(s(x), s(y)) → EQ(x, y)
The graph contains the following edges 1 > 1, 2 > 2
(20) YES
(21) 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.
(22) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(23) 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:
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
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.
(24) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
or(true, x0)
or(false, x0)
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)
(25) 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:
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
The set Q consists of the following terms:
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
union(empty, x0)
union(edge(x0, x1, x2), x3)
We have to consider all minimal (P,Q,R)-chains.
(26) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
IF_REACH_1(true, x, y, edge(u, v, i), h) → IF_REACH_2(eq(y, v), x, y, edge(u, v, i), h)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( IF_REACH_1(x1, ..., x5) ) = 2x1 + 2x4 + 2x5 + 2 |
POL( IF_REACH_2(x1, ..., x5) ) = 2x4 + 2x5 |
POL( REACH(x1, ..., x4) ) = 2x3 + 2x4 + 2 |
POL( union(x1, x2) ) = x1 + x2 |
POL( edge(x1, ..., x3) ) = x3 + 1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
(27) 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_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:
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
The set Q consists of the following terms:
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
union(empty, x0)
union(edge(x0, x1, x2), x3)
We have to consider all minimal (P,Q,R)-chains.
(28) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(29) 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:
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
The set Q consists of the following terms:
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
union(empty, x0)
union(edge(x0, x1, x2), x3)
We have to consider all minimal (P,Q,R)-chains.
(30) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(31) 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)
The set Q consists of the following terms:
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
union(empty, x0)
union(edge(x0, x1, x2), x3)
We have to consider all minimal (P,Q,R)-chains.
(32) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
union(empty, x0)
union(edge(x0, x1, x2), x3)
(33) 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)
The set Q consists of the following terms:
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(34) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- REACH(x, y, edge(u, v, i), h) → IF_REACH_1(eq(x, u), x, y, edge(u, v, i), h)
The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5
- IF_REACH_1(false, x, y, edge(u, v, i), h) → REACH(x, y, i, edge(u, v, h))
The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 3
(35) YES