YES Termination w.r.t. Q proof of AProVE_07_thiemann37.ari

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

0 QTRS
↳1 Overlay + Local Confluence (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 21 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 QDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 QDP
        ↳10 QReductionProof (⇔, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 QDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 QDP
        ↳17 QReductionProof (⇔, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 QDP
        ↳22 UsableRulesProof (⇔, 0 ms)
        ↳23 QDP
        ↳24 QReductionProof (⇔, 0 ms)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔, 0 ms)
        ↳27 YES
    ↳28 QDP
        ↳29 UsableRulesProof (⇔, 0 ms)
        ↳30 QDP
        ↳31 QReductionProof (⇔, 0 ms)
        ↳32 QDP
        ↳33 TransformationProof (⇔, 0 ms)
        ↳34 QDP
        ↳35 TransformationProof (⇔, 0 ms)
        ↳36 QDP
        ↳37 QDPQMonotonicMRRProof (⇔, 0 ms)
        ↳38 QDP
        ↳39 QDPQMonotonicMRRProof (⇔, 13 ms)
        ↳40 QDP
        ↳41 NonInfProof (⇔, 19 ms)
        ↳42 QDP
        ↳43 DependencyGraphProof (⇔, 0 ms)
        ↳44 QDP
        ↳45 UsableRulesProof (⇔, 0 ms)
        ↳46 QDP
        ↳47 QReductionProof (⇔, 0 ms)
        ↳48 QDP
        ↳49 QDPSizeChangeProof (⇔, 0 ms)
        ↳50 YES

(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
and(true, y) → y
and(false, y) → false
size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0, i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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
and(true, y) → y
and(false, y) → false
size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0, i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

(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)
SIZE(edge(x, y, i)) → SIZE(i)
LE(s(x), s(y)) → LE(x, y)
REACHABLE(x, y, i) → REACH(x, y, 0, i, i)
REACH(x, y, c, i, j) → IF1(eq(x, y), x, y, c, i, j)
REACH(x, y, c, i, j) → EQ(x, y)
IF1(false, x, y, c, i, j) → IF2(le(c, size(j)), x, y, c, i, j)
IF1(false, x, y, c, i, j) → LE(c, size(j))
IF1(false, x, y, c, i, j) → SIZE(j)
IF2(true, x, y, c, edge(u, v, i), j) → OR(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))
IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → AND(eq(x, u), reach(v, y, s(c), j, j))
IF2(true, x, y, c, edge(u, v, i), j) → EQ(x, u)
IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)

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
and(true, y) → y
and(false, y) → false
size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0, i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LE(s(x), s(y)) → LE(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
and(true, y) → y
and(false, y) → false
size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0, i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

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:

LE(s(x), s(y)) → LE(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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

(11) Obligation:

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

LE(s(x), s(y)) → LE(x, y)

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:

  • LE(s(x), s(y)) → LE(x, y)
    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:

SIZE(edge(x, y, i)) → SIZE(i)

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
and(true, y) → y
and(false, y) → false
size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0, i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

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:

SIZE(edge(x, y, i)) → SIZE(i)

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

(18) Obligation:

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

SIZE(edge(x, y, i)) → SIZE(i)

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:

  • SIZE(edge(x, y, i)) → SIZE(i)
    The graph contains the following edges 1 > 1

(20) YES

(21) 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
and(true, y) → y
and(false, y) → false
size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0, i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

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:

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

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].

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

(25) 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.

(26) 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

(27) YES

(28) Obligation:

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

REACH(x, y, c, i, j) → IF1(eq(x, y), x, y, c, i, j)
IF1(false, x, y, c, i, j) → IF2(le(c, size(j)), x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)

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
and(true, y) → y
and(false, y) → false
size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0, i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

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

(29) 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.

(30) Obligation:

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

REACH(x, y, c, i, j) → IF1(eq(x, y), x, y, c, i, j)
IF1(false, x, y, c, i, j) → IF2(le(c, size(j)), x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)

The TRS R consists of the following rules:

size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
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)
and(true, x0)
and(false, x0)
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

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

(31) 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)
and(true, x0)
and(false, x0)
reachable(x0, x1, x2)
reach(x0, x1, x2, x3, x4)
if1(true, x0, x1, x2, x3, x4)
if1(false, x0, x1, x2, x3, x4)
if2(false, x0, x1, x2, x3, x4)
if2(true, x0, x1, x2, empty, x3)
if2(true, x0, x1, x2, edge(x3, x4, x5), x6)

(32) Obligation:

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

REACH(x, y, c, i, j) → IF1(eq(x, y), x, y, c, i, j)
IF1(false, x, y, c, i, j) → IF2(le(c, size(j)), x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)

The TRS R consists of the following rules:

size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
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))
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

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

(33) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REACH(x, y, c, i, j) → IF1(eq(x, y), x, y, c, i, j) we obtained the following new rules [LPAR04]:

REACH(z4, z1, s(z2), z6, z6) → IF1(eq(z4, z1), z4, z1, s(z2), z6, z6) → REACH(z4, z1, s(z2), z6, z6) → IF1(eq(z4, z1), z4, z1, s(z2), z6, z6)

(34) Obligation:

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

IF1(false, x, y, c, i, j) → IF2(le(c, size(j)), x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)
REACH(z4, z1, s(z2), z6, z6) → IF1(eq(z4, z1), z4, z1, s(z2), z6, z6)

The TRS R consists of the following rules:

size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
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))
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

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

(35) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IF1(false, x, y, c, i, j) → IF2(le(c, size(j)), x, y, c, i, j) we obtained the following new rules [LPAR04]:

IF1(false, z0, z1, s(z2), z3, z3) → IF2(le(s(z2), size(z3)), z0, z1, s(z2), z3, z3) → IF1(false, z0, z1, s(z2), z3, z3) → IF2(le(s(z2), size(z3)), z0, z1, s(z2), z3, z3)

(36) Obligation:

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

IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)
REACH(z4, z1, s(z2), z6, z6) → IF1(eq(z4, z1), z4, z1, s(z2), z6, z6)
IF1(false, z0, z1, s(z2), z3, z3) → IF2(le(s(z2), size(z3)), z0, z1, s(z2), z3, z3)

The TRS R consists of the following rules:

size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
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))
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

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

(37) QDPQMonotonicMRRProof (EQUIVALENT transformation)

By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented rules of the TRS R:

eq(0, 0) → true

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(IF1(x1, x2, x3, x4, x5, x6)) = x1 + x4   
POL(IF2(x1, x2, x3, x4, x5, x6)) = 2   
POL(REACH(x1, x2, x3, x4, x5)) = 2·x3   
POL(edge(x1, x2, x3)) = 2 + 2·x3   
POL(empty) = 0   
POL(eq(x1, x2)) = 1   
POL(false) = 1   
POL(le(x1, x2)) = 2   
POL(s(x1)) = 1   
POL(size(x1)) = 2 + 2·x1   
POL(true) = 0   

(38) Obligation:

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

IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)
REACH(z4, z1, s(z2), z6, z6) → IF1(eq(z4, z1), z4, z1, s(z2), z6, z6)
IF1(false, z0, z1, s(z2), z3, z3) → IF2(le(s(z2), size(z3)), z0, z1, s(z2), z3, z3)

The TRS R consists of the following rules:

size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
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))
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

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

(39) QDPQMonotonicMRRProof (EQUIVALENT transformation)

By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented rules of the TRS R:

le(s(x), 0) → false

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(IF1(x1, x2, x3, x4, x5, x6)) = 2 + 2·x6   
POL(IF2(x1, x2, x3, x4, x5, x6)) = 2·x1 + 2·x6   
POL(REACH(x1, x2, x3, x4, x5)) = 2 + 2·x5   
POL(edge(x1, x2, x3)) = 2·x3   
POL(empty) = 0   
POL(eq(x1, x2)) = 0   
POL(false) = 0   
POL(le(x1, x2)) = 1 + x2   
POL(s(x1)) = 2·x1   
POL(size(x1)) = 0   
POL(true) = 1   

(40) Obligation:

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

IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)
REACH(z4, z1, s(z2), z6, z6) → IF1(eq(z4, z1), z4, z1, s(z2), z6, z6)
IF1(false, z0, z1, s(z2), z3, z3) → IF2(le(s(z2), size(z3)), z0, z1, s(z2), z3, z3)

The TRS R consists of the following rules:

size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), s(y)) → le(x, y)
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))
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

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

(41) NonInfProof (EQUIVALENT transformation)

The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j) the following chains were created:
  • We consider the chain IF2(true, x0, x1, x2, edge(x3, x4, x5), x6) → IF2(true, x0, x1, x2, x5, x6), IF2(true, x7, x8, x9, edge(x10, x11, x12), x13) → IF2(true, x7, x8, x9, x12, x13) which results in the following constraint:
    (1)    (IF2(true, x0, x1, x2, x5, x6)=IF2(true, x7, x8, x9, edge(x10, x11, x12), x13) ⇒ IF2(true, x0, x1, x2, edge(x3, x4, x5), x6)≥IF2(true, x0, x1, x2, x5, x6))


    We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
    (2)    (IF2(true, x0, x1, x2, edge(x3, x4, edge(x10, x11, x12)), x6)≥IF2(true, x0, x1, x2, edge(x10, x11, x12), x6))


  • We consider the chain IF2(true, x14, x15, x16, edge(x17, x18, x19), x20) → IF2(true, x14, x15, x16, x19, x20), IF2(true, x21, x22, x23, edge(x24, x25, x26), x27) → REACH(x25, x22, s(x23), x27, x27) which results in the following constraint:
    (1)    (IF2(true, x14, x15, x16, x19, x20)=IF2(true, x21, x22, x23, edge(x24, x25, x26), x27) ⇒ IF2(true, x14, x15, x16, edge(x17, x18, x19), x20)≥IF2(true, x14, x15, x16, x19, x20))


    We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
    (2)    (IF2(true, x14, x15, x16, edge(x17, x18, edge(x24, x25, x26)), x20)≥IF2(true, x14, x15, x16, edge(x24, x25, x26), x20))






For Pair IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j) the following chains were created:
  • We consider the chain IF2(true, x56, x57, x58, edge(x59, x60, x61), x62) → REACH(x60, x57, s(x58), x62, x62), REACH(x63, x64, s(x65), x66, x66) → IF1(eq(x63, x64), x63, x64, s(x65), x66, x66) which results in the following constraint:
    (1)    (REACH(x60, x57, s(x58), x62, x62)=REACH(x63, x64, s(x65), x66, x66) ⇒ IF2(true, x56, x57, x58, edge(x59, x60, x61), x62)≥REACH(x60, x57, s(x58), x62, x62))


    We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
    (2)    (IF2(true, x56, x57, x58, edge(x59, x60, x61), x62)≥REACH(x60, x57, s(x58), x62, x62))






For Pair REACH(z4, z1, s(z2), z6, z6) → IF1(eq(z4, z1), z4, z1, s(z2), z6, z6) the following chains were created:
  • We consider the chain REACH(x86, x87, s(x88), x89, x89) → IF1(eq(x86, x87), x86, x87, s(x88), x89, x89), IF1(false, x90, x91, s(x92), x93, x93) → IF2(le(s(x92), size(x93)), x90, x91, s(x92), x93, x93) which results in the following constraint:
    (1)    (IF1(eq(x86, x87), x86, x87, s(x88), x89, x89)=IF1(false, x90, x91, s(x92), x93, x93) ⇒ REACH(x86, x87, s(x88), x89, x89)≥IF1(eq(x86, x87), x86, x87, s(x88), x89, x89))


    We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
    (2)    (eq(x86, x87)=false ⇒ REACH(x86, x87, s(x88), x89, x89)≥IF1(eq(x86, x87), x86, x87, s(x88), x89, x89))


    We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on eq(x86, x87)=false which results in the following new constraints:
    (3)    (false=false ⇒ REACH(0, s(x124), s(x88), x89, x89)≥IF1(eq(0, s(x124)), 0, s(x124), s(x88), x89, x89))

    (4)    (false=false ⇒ REACH(s(x125), 0, s(x88), x89, x89)≥IF1(eq(s(x125), 0), s(x125), 0, s(x88), x89, x89))

    (5)    (eq(x127, x126)=false∧(∀x128,x129:eq(x127, x126)=false ⇒ REACH(x127, x126, s(x128), x129, x129)≥IF1(eq(x127, x126), x127, x126, s(x128), x129, x129)) ⇒ REACH(s(x127), s(x126), s(x88), x89, x89)≥IF1(eq(s(x127), s(x126)), s(x127), s(x126), s(x88), x89, x89))


    We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
    (6)    (REACH(0, s(x124), s(x88), x89, x89)≥IF1(eq(0, s(x124)), 0, s(x124), s(x88), x89, x89))


    We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
    (7)    (REACH(s(x125), 0, s(x88), x89, x89)≥IF1(eq(s(x125), 0), s(x125), 0, s(x88), x89, x89))


    We simplified constraint (5) using rule (VI) where we applied the induction hypothesis (∀x128,x129:eq(x127, x126)=false ⇒ REACH(x127, x126, s(x128), x129, x129)≥IF1(eq(x127, x126), x127, x126, s(x128), x129, x129)) with σ = [x128 / x88, x129 / x89] which results in the following new constraint:
    (8)    (REACH(x127, x126, s(x88), x89, x89)≥IF1(eq(x127, x126), x127, x126, s(x88), x89, x89) ⇒ REACH(s(x127), s(x126), s(x88), x89, x89)≥IF1(eq(s(x127), s(x126)), s(x127), s(x126), s(x88), x89, x89))






For Pair IF1(false, z0, z1, s(z2), z3, z3) → IF2(le(s(z2), size(z3)), z0, z1, s(z2), z3, z3) the following chains were created:
  • We consider the chain IF1(false, x94, x95, s(x96), x97, x97) → IF2(le(s(x96), size(x97)), x94, x95, s(x96), x97, x97), IF2(true, x98, x99, x100, edge(x101, x102, x103), x104) → IF2(true, x98, x99, x100, x103, x104) which results in the following constraint:
    (1)    (IF2(le(s(x96), size(x97)), x94, x95, s(x96), x97, x97)=IF2(true, x98, x99, x100, edge(x101, x102, x103), x104) ⇒ IF1(false, x94, x95, s(x96), x97, x97)≥IF2(le(s(x96), size(x97)), x94, x95, s(x96), x97, x97))


    We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
    (2)    (s(x96)=x130∧edge(x101, x102, x103)=x132∧size(x132)=x131∧le(x130, x131)=true ⇒ IF1(false, x94, x95, s(x96), edge(x101, x102, x103), edge(x101, x102, x103))≥IF2(le(s(x96), size(edge(x101, x102, x103))), x94, x95, s(x96), edge(x101, x102, x103), edge(x101, x102, x103)))


    We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on le(x130, x131)=true which results in the following new constraints:
    (3)    (true=true∧s(x96)=0∧edge(x101, x102, x103)=x132∧size(x132)=x133 ⇒ IF1(false, x94, x95, s(x96), edge(x101, x102, x103), edge(x101, x102, x103))≥IF2(le(s(x96), size(edge(x101, x102, x103))), x94, x95, s(x96), edge(x101, x102, x103), edge(x101, x102, x103)))

    (4)    (le(x135, x134)=true∧s(x96)=s(x135)∧edge(x101, x102, x103)=x132∧size(x132)=s(x134)∧(∀x136,x137,x138,x139,x140,x141,x142:le(x135, x134)=true∧s(x136)=x135∧edge(x137, x138, x139)=x140∧size(x140)=x134 ⇒ IF1(false, x141, x142, s(x136), edge(x137, x138, x139), edge(x137, x138, x139))≥IF2(le(s(x136), size(edge(x137, x138, x139))), x141, x142, s(x136), edge(x137, x138, x139), edge(x137, x138, x139))) ⇒ IF1(false, x94, x95, s(x96), edge(x101, x102, x103), edge(x101, x102, x103))≥IF2(le(s(x96), size(edge(x101, x102, x103))), x94, x95, s(x96), edge(x101, x102, x103), edge(x101, x102, x103)))


    We solved constraint (3) using rules (I), (II).We simplified constraint (4) using rules (I), (II), (III) which results in the following new constraint:
    (5)    (le(x135, x134)=true∧edge(x101, x102, x103)=x132∧size(x132)=s(x134)∧(∀x136,x137,x138,x139,x140,x141,x142:le(x135, x134)=true∧s(x136)=x135∧edge(x137, x138, x139)=x140∧size(x140)=x134 ⇒ IF1(false, x141, x142, s(x136), edge(x137, x138, x139), edge(x137, x138, x139))≥IF2(le(s(x136), size(edge(x137, x138, x139))), x141, x142, s(x136), edge(x137, x138, x139), edge(x137, x138, x139))) ⇒ IF1(false, x94, x95, s(x135), edge(x101, x102, x103), edge(x101, x102, x103))≥IF2(le(s(x135), size(edge(x101, x102, x103))), x94, x95, s(x135), edge(x101, x102, x103), edge(x101, x102, x103)))


    We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on size(x132)=s(x134) which results in the following new constraint:
    (6)    (s(size(x143))=s(x134)∧le(x135, x134)=true∧edge(x101, x102, x103)=edge(x145, x144, x143)∧(∀x136,x137,x138,x139,x140,x141,x142:le(x135, x134)=true∧s(x136)=x135∧edge(x137, x138, x139)=x140∧size(x140)=x134 ⇒ IF1(false, x141, x142, s(x136), edge(x137, x138, x139), edge(x137, x138, x139))≥IF2(le(s(x136), size(edge(x137, x138, x139))), x141, x142, s(x136), edge(x137, x138, x139), edge(x137, x138, x139)))∧(∀x146,x147,x148,x149,x150,x151,x152,x153,x154,x155,x156,x157,x158,x159:size(x143)=s(x146)∧le(x147, x146)=true∧edge(x148, x149, x150)=x143∧(∀x151,x152,x153,x154,x155,x156,x157:le(x147, x146)=true∧s(x151)=x147∧edge(x152, x153, x154)=x155∧size(x155)=x146 ⇒ IF1(false, x156, x157, s(x151), edge(x152, x153, x154), edge(x152, x153, x154))≥IF2(le(s(x151), size(edge(x152, x153, x154))), x156, x157, s(x151), edge(x152, x153, x154), edge(x152, x153, x154))) ⇒ IF1(false, x158, x159, s(x147), edge(x148, x149, x150), edge(x148, x149, x150))≥IF2(le(s(x147), size(edge(x148, x149, x150))), x158, x159, s(x147), edge(x148, x149, x150), edge(x148, x149, x150))) ⇒ IF1(false, x94, x95, s(x135), edge(x101, x102, x103), edge(x101, x102, x103))≥IF2(le(s(x135), size(edge(x101, x102, x103))), x94, x95, s(x135), edge(x101, x102, x103), edge(x101, x102, x103)))


    We simplified constraint (6) using rules (I), (II), (III), (IV) which results in the following new constraint:
    (7)    (size(x143)=x134∧le(x135, x134)=true ⇒ IF1(false, x94, x95, s(x135), edge(x101, x102, x143), edge(x101, x102, x143))≥IF2(le(s(x135), size(edge(x101, x102, x143))), x94, x95, s(x135), edge(x101, x102, x143), edge(x101, x102, x143)))


    We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on le(x135, x134)=true which results in the following new constraints:
    (8)    (true=true∧size(x143)=x160 ⇒ IF1(false, x94, x95, s(0), edge(x101, x102, x143), edge(x101, x102, x143))≥IF2(le(s(0), size(edge(x101, x102, x143))), x94, x95, s(0), edge(x101, x102, x143), edge(x101, x102, x143)))

    (9)    (le(x162, x161)=true∧size(x143)=s(x161)∧(∀x163,x164,x165,x166,x167:le(x162, x161)=true∧size(x163)=x161 ⇒ IF1(false, x164, x165, s(x162), edge(x166, x167, x163), edge(x166, x167, x163))≥IF2(le(s(x162), size(edge(x166, x167, x163))), x164, x165, s(x162), edge(x166, x167, x163), edge(x166, x167, x163))) ⇒ IF1(false, x94, x95, s(s(x162)), edge(x101, x102, x143), edge(x101, x102, x143))≥IF2(le(s(s(x162)), size(edge(x101, x102, x143))), x94, x95, s(s(x162)), edge(x101, x102, x143), edge(x101, x102, x143)))


    We simplified constraint (8) using rules (I), (II), (IV) which results in the following new constraint:
    (10)    (IF1(false, x94, x95, s(0), edge(x101, x102, x143), edge(x101, x102, x143))≥IF2(le(s(0), size(edge(x101, x102, x143))), x94, x95, s(0), edge(x101, x102, x143), edge(x101, x102, x143)))


    We simplified constraint (9) using rule (V) (with possible (I) afterwards) using induction on size(x143)=s(x161) which results in the following new constraint:
    (11)    (s(size(x168))=s(x161)∧le(x162, x161)=true∧(∀x163,x164,x165,x166,x167:le(x162, x161)=true∧size(x163)=x161 ⇒ IF1(false, x164, x165, s(x162), edge(x166, x167, x163), edge(x166, x167, x163))≥IF2(le(s(x162), size(edge(x166, x167, x163))), x164, x165, s(x162), edge(x166, x167, x163), edge(x166, x167, x163)))∧(∀x171,x172,x173,x174,x175,x176,x177,x178,x179,x180,x181:size(x168)=s(x171)∧le(x172, x171)=true∧(∀x173,x174,x175,x176,x177:le(x172, x171)=true∧size(x173)=x171 ⇒ IF1(false, x174, x175, s(x172), edge(x176, x177, x173), edge(x176, x177, x173))≥IF2(le(s(x172), size(edge(x176, x177, x173))), x174, x175, s(x172), edge(x176, x177, x173), edge(x176, x177, x173))) ⇒ IF1(false, x178, x179, s(s(x172)), edge(x180, x181, x168), edge(x180, x181, x168))≥IF2(le(s(s(x172)), size(edge(x180, x181, x168))), x178, x179, s(s(x172)), edge(x180, x181, x168), edge(x180, x181, x168))) ⇒ IF1(false, x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168)))≥IF2(le(s(s(x162)), size(edge(x101, x102, edge(x170, x169, x168)))), x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168))))


    We simplified constraint (11) using rules (I), (II) which results in the following new constraint:
    (12)    (size(x168)=x161∧le(x162, x161)=true∧(∀x163,x164,x165,x166,x167:le(x162, x161)=true∧size(x163)=x161 ⇒ IF1(false, x164, x165, s(x162), edge(x166, x167, x163), edge(x166, x167, x163))≥IF2(le(s(x162), size(edge(x166, x167, x163))), x164, x165, s(x162), edge(x166, x167, x163), edge(x166, x167, x163)))∧(∀x171,x172,x173,x174,x175,x176,x177,x178,x179,x180,x181:size(x168)=s(x171)∧le(x172, x171)=true∧(∀x173,x174,x175,x176,x177:le(x172, x171)=true∧size(x173)=x171 ⇒ IF1(false, x174, x175, s(x172), edge(x176, x177, x173), edge(x176, x177, x173))≥IF2(le(s(x172), size(edge(x176, x177, x173))), x174, x175, s(x172), edge(x176, x177, x173), edge(x176, x177, x173))) ⇒ IF1(false, x178, x179, s(s(x172)), edge(x180, x181, x168), edge(x180, x181, x168))≥IF2(le(s(s(x172)), size(edge(x180, x181, x168))), x178, x179, s(s(x172)), edge(x180, x181, x168), edge(x180, x181, x168))) ⇒ IF1(false, x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168)))≥IF2(le(s(s(x162)), size(edge(x101, x102, edge(x170, x169, x168)))), x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168))))


    We simplified constraint (12) using rule (VI) where we applied the induction hypothesis (∀x163,x164,x165,x166,x167:le(x162, x161)=true∧size(x163)=x161 ⇒ IF1(false, x164, x165, s(x162), edge(x166, x167, x163), edge(x166, x167, x163))≥IF2(le(s(x162), size(edge(x166, x167, x163))), x164, x165, s(x162), edge(x166, x167, x163), edge(x166, x167, x163))) with σ = [x163 / x168, x164 / x94, x165 / x95, x166 / x101, x167 / x102] which results in the following new constraint:
    (13)    (IF1(false, x94, x95, s(x162), edge(x101, x102, x168), edge(x101, x102, x168))≥IF2(le(s(x162), size(edge(x101, x102, x168))), x94, x95, s(x162), edge(x101, x102, x168), edge(x101, x102, x168))∧(∀x171,x172,x173,x174,x175,x176,x177,x178,x179,x180,x181:size(x168)=s(x171)∧le(x172, x171)=true∧(∀x173,x174,x175,x176,x177:le(x172, x171)=true∧size(x173)=x171 ⇒ IF1(false, x174, x175, s(x172), edge(x176, x177, x173), edge(x176, x177, x173))≥IF2(le(s(x172), size(edge(x176, x177, x173))), x174, x175, s(x172), edge(x176, x177, x173), edge(x176, x177, x173))) ⇒ IF1(false, x178, x179, s(s(x172)), edge(x180, x181, x168), edge(x180, x181, x168))≥IF2(le(s(s(x172)), size(edge(x180, x181, x168))), x178, x179, s(s(x172)), edge(x180, x181, x168), edge(x180, x181, x168))) ⇒ IF1(false, x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168)))≥IF2(le(s(s(x162)), size(edge(x101, x102, edge(x170, x169, x168)))), x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168))))


    We simplified constraint (13) using rule (IV) which results in the following new constraint:
    (14)    (IF1(false, x94, x95, s(x162), edge(x101, x102, x168), edge(x101, x102, x168))≥IF2(le(s(x162), size(edge(x101, x102, x168))), x94, x95, s(x162), edge(x101, x102, x168), edge(x101, x102, x168)) ⇒ IF1(false, x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168)))≥IF2(le(s(s(x162)), size(edge(x101, x102, edge(x170, x169, x168)))), x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168))))


  • We consider the chain IF1(false, x105, x106, s(x107), x108, x108) → IF2(le(s(x107), size(x108)), x105, x106, s(x107), x108, x108), IF2(true, x109, x110, x111, edge(x112, x113, x114), x115) → REACH(x113, x110, s(x111), x115, x115) which results in the following constraint:
    (1)    (IF2(le(s(x107), size(x108)), x105, x106, s(x107), x108, x108)=IF2(true, x109, x110, x111, edge(x112, x113, x114), x115) ⇒ IF1(false, x105, x106, s(x107), x108, x108)≥IF2(le(s(x107), size(x108)), x105, x106, s(x107), x108, x108))


    We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
    (2)    (s(x107)=x182∧edge(x112, x113, x114)=x184∧size(x184)=x183∧le(x182, x183)=true ⇒ IF1(false, x105, x106, s(x107), edge(x112, x113, x114), edge(x112, x113, x114))≥IF2(le(s(x107), size(edge(x112, x113, x114))), x105, x106, s(x107), edge(x112, x113, x114), edge(x112, x113, x114)))


    We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on le(x182, x183)=true which results in the following new constraints:
    (3)    (true=true∧s(x107)=0∧edge(x112, x113, x114)=x184∧size(x184)=x185 ⇒ IF1(false, x105, x106, s(x107), edge(x112, x113, x114), edge(x112, x113, x114))≥IF2(le(s(x107), size(edge(x112, x113, x114))), x105, x106, s(x107), edge(x112, x113, x114), edge(x112, x113, x114)))

    (4)    (le(x187, x186)=true∧s(x107)=s(x187)∧edge(x112, x113, x114)=x184∧size(x184)=s(x186)∧(∀x188,x189,x190,x191,x192,x193,x194:le(x187, x186)=true∧s(x188)=x187∧edge(x189, x190, x191)=x192∧size(x192)=x186 ⇒ IF1(false, x193, x194, s(x188), edge(x189, x190, x191), edge(x189, x190, x191))≥IF2(le(s(x188), size(edge(x189, x190, x191))), x193, x194, s(x188), edge(x189, x190, x191), edge(x189, x190, x191))) ⇒ IF1(false, x105, x106, s(x107), edge(x112, x113, x114), edge(x112, x113, x114))≥IF2(le(s(x107), size(edge(x112, x113, x114))), x105, x106, s(x107), edge(x112, x113, x114), edge(x112, x113, x114)))


    We solved constraint (3) using rules (I), (II).We simplified constraint (4) using rules (I), (II), (III) which results in the following new constraint:
    (5)    (le(x187, x186)=true∧edge(x112, x113, x114)=x184∧size(x184)=s(x186)∧(∀x188,x189,x190,x191,x192,x193,x194:le(x187, x186)=true∧s(x188)=x187∧edge(x189, x190, x191)=x192∧size(x192)=x186 ⇒ IF1(false, x193, x194, s(x188), edge(x189, x190, x191), edge(x189, x190, x191))≥IF2(le(s(x188), size(edge(x189, x190, x191))), x193, x194, s(x188), edge(x189, x190, x191), edge(x189, x190, x191))) ⇒ IF1(false, x105, x106, s(x187), edge(x112, x113, x114), edge(x112, x113, x114))≥IF2(le(s(x187), size(edge(x112, x113, x114))), x105, x106, s(x187), edge(x112, x113, x114), edge(x112, x113, x114)))


    We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on size(x184)=s(x186) which results in the following new constraint:
    (6)    (s(size(x195))=s(x186)∧le(x187, x186)=true∧edge(x112, x113, x114)=edge(x197, x196, x195)∧(∀x188,x189,x190,x191,x192,x193,x194:le(x187, x186)=true∧s(x188)=x187∧edge(x189, x190, x191)=x192∧size(x192)=x186 ⇒ IF1(false, x193, x194, s(x188), edge(x189, x190, x191), edge(x189, x190, x191))≥IF2(le(s(x188), size(edge(x189, x190, x191))), x193, x194, s(x188), edge(x189, x190, x191), edge(x189, x190, x191)))∧(∀x198,x199,x200,x201,x202,x203,x204,x205,x206,x207,x208,x209,x210,x211:size(x195)=s(x198)∧le(x199, x198)=true∧edge(x200, x201, x202)=x195∧(∀x203,x204,x205,x206,x207,x208,x209:le(x199, x198)=true∧s(x203)=x199∧edge(x204, x205, x206)=x207∧size(x207)=x198 ⇒ IF1(false, x208, x209, s(x203), edge(x204, x205, x206), edge(x204, x205, x206))≥IF2(le(s(x203), size(edge(x204, x205, x206))), x208, x209, s(x203), edge(x204, x205, x206), edge(x204, x205, x206))) ⇒ IF1(false, x210, x211, s(x199), edge(x200, x201, x202), edge(x200, x201, x202))≥IF2(le(s(x199), size(edge(x200, x201, x202))), x210, x211, s(x199), edge(x200, x201, x202), edge(x200, x201, x202))) ⇒ IF1(false, x105, x106, s(x187), edge(x112, x113, x114), edge(x112, x113, x114))≥IF2(le(s(x187), size(edge(x112, x113, x114))), x105, x106, s(x187), edge(x112, x113, x114), edge(x112, x113, x114)))


    We simplified constraint (6) using rules (I), (II), (III), (IV) which results in the following new constraint:
    (7)    (size(x195)=x186∧le(x187, x186)=true ⇒ IF1(false, x105, x106, s(x187), edge(x112, x113, x195), edge(x112, x113, x195))≥IF2(le(s(x187), size(edge(x112, x113, x195))), x105, x106, s(x187), edge(x112, x113, x195), edge(x112, x113, x195)))


    We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on le(x187, x186)=true which results in the following new constraints:
    (8)    (true=true∧size(x195)=x212 ⇒ IF1(false, x105, x106, s(0), edge(x112, x113, x195), edge(x112, x113, x195))≥IF2(le(s(0), size(edge(x112, x113, x195))), x105, x106, s(0), edge(x112, x113, x195), edge(x112, x113, x195)))

    (9)    (le(x214, x213)=true∧size(x195)=s(x213)∧(∀x215,x216,x217,x218,x219:le(x214, x213)=true∧size(x215)=x213 ⇒ IF1(false, x216, x217, s(x214), edge(x218, x219, x215), edge(x218, x219, x215))≥IF2(le(s(x214), size(edge(x218, x219, x215))), x216, x217, s(x214), edge(x218, x219, x215), edge(x218, x219, x215))) ⇒ IF1(false, x105, x106, s(s(x214)), edge(x112, x113, x195), edge(x112, x113, x195))≥IF2(le(s(s(x214)), size(edge(x112, x113, x195))), x105, x106, s(s(x214)), edge(x112, x113, x195), edge(x112, x113, x195)))


    We simplified constraint (8) using rules (I), (II), (IV) which results in the following new constraint:
    (10)    (IF1(false, x105, x106, s(0), edge(x112, x113, x195), edge(x112, x113, x195))≥IF2(le(s(0), size(edge(x112, x113, x195))), x105, x106, s(0), edge(x112, x113, x195), edge(x112, x113, x195)))


    We simplified constraint (9) using rule (V) (with possible (I) afterwards) using induction on size(x195)=s(x213) which results in the following new constraint:
    (11)    (s(size(x220))=s(x213)∧le(x214, x213)=true∧(∀x215,x216,x217,x218,x219:le(x214, x213)=true∧size(x215)=x213 ⇒ IF1(false, x216, x217, s(x214), edge(x218, x219, x215), edge(x218, x219, x215))≥IF2(le(s(x214), size(edge(x218, x219, x215))), x216, x217, s(x214), edge(x218, x219, x215), edge(x218, x219, x215)))∧(∀x223,x224,x225,x226,x227,x228,x229,x230,x231,x232,x233:size(x220)=s(x223)∧le(x224, x223)=true∧(∀x225,x226,x227,x228,x229:le(x224, x223)=true∧size(x225)=x223 ⇒ IF1(false, x226, x227, s(x224), edge(x228, x229, x225), edge(x228, x229, x225))≥IF2(le(s(x224), size(edge(x228, x229, x225))), x226, x227, s(x224), edge(x228, x229, x225), edge(x228, x229, x225))) ⇒ IF1(false, x230, x231, s(s(x224)), edge(x232, x233, x220), edge(x232, x233, x220))≥IF2(le(s(s(x224)), size(edge(x232, x233, x220))), x230, x231, s(s(x224)), edge(x232, x233, x220), edge(x232, x233, x220))) ⇒ IF1(false, x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220)))≥IF2(le(s(s(x214)), size(edge(x112, x113, edge(x222, x221, x220)))), x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220))))


    We simplified constraint (11) using rules (I), (II) which results in the following new constraint:
    (12)    (size(x220)=x213∧le(x214, x213)=true∧(∀x215,x216,x217,x218,x219:le(x214, x213)=true∧size(x215)=x213 ⇒ IF1(false, x216, x217, s(x214), edge(x218, x219, x215), edge(x218, x219, x215))≥IF2(le(s(x214), size(edge(x218, x219, x215))), x216, x217, s(x214), edge(x218, x219, x215), edge(x218, x219, x215)))∧(∀x223,x224,x225,x226,x227,x228,x229,x230,x231,x232,x233:size(x220)=s(x223)∧le(x224, x223)=true∧(∀x225,x226,x227,x228,x229:le(x224, x223)=true∧size(x225)=x223 ⇒ IF1(false, x226, x227, s(x224), edge(x228, x229, x225), edge(x228, x229, x225))≥IF2(le(s(x224), size(edge(x228, x229, x225))), x226, x227, s(x224), edge(x228, x229, x225), edge(x228, x229, x225))) ⇒ IF1(false, x230, x231, s(s(x224)), edge(x232, x233, x220), edge(x232, x233, x220))≥IF2(le(s(s(x224)), size(edge(x232, x233, x220))), x230, x231, s(s(x224)), edge(x232, x233, x220), edge(x232, x233, x220))) ⇒ IF1(false, x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220)))≥IF2(le(s(s(x214)), size(edge(x112, x113, edge(x222, x221, x220)))), x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220))))


    We simplified constraint (12) using rule (VI) where we applied the induction hypothesis (∀x215,x216,x217,x218,x219:le(x214, x213)=true∧size(x215)=x213 ⇒ IF1(false, x216, x217, s(x214), edge(x218, x219, x215), edge(x218, x219, x215))≥IF2(le(s(x214), size(edge(x218, x219, x215))), x216, x217, s(x214), edge(x218, x219, x215), edge(x218, x219, x215))) with σ = [x215 / x220, x216 / x105, x217 / x106, x218 / x112, x219 / x113] which results in the following new constraint:
    (13)    (IF1(false, x105, x106, s(x214), edge(x112, x113, x220), edge(x112, x113, x220))≥IF2(le(s(x214), size(edge(x112, x113, x220))), x105, x106, s(x214), edge(x112, x113, x220), edge(x112, x113, x220))∧(∀x223,x224,x225,x226,x227,x228,x229,x230,x231,x232,x233:size(x220)=s(x223)∧le(x224, x223)=true∧(∀x225,x226,x227,x228,x229:le(x224, x223)=true∧size(x225)=x223 ⇒ IF1(false, x226, x227, s(x224), edge(x228, x229, x225), edge(x228, x229, x225))≥IF2(le(s(x224), size(edge(x228, x229, x225))), x226, x227, s(x224), edge(x228, x229, x225), edge(x228, x229, x225))) ⇒ IF1(false, x230, x231, s(s(x224)), edge(x232, x233, x220), edge(x232, x233, x220))≥IF2(le(s(s(x224)), size(edge(x232, x233, x220))), x230, x231, s(s(x224)), edge(x232, x233, x220), edge(x232, x233, x220))) ⇒ IF1(false, x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220)))≥IF2(le(s(s(x214)), size(edge(x112, x113, edge(x222, x221, x220)))), x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220))))


    We simplified constraint (13) using rule (IV) which results in the following new constraint:
    (14)    (IF1(false, x105, x106, s(x214), edge(x112, x113, x220), edge(x112, x113, x220))≥IF2(le(s(x214), size(edge(x112, x113, x220))), x105, x106, s(x214), edge(x112, x113, x220), edge(x112, x113, x220)) ⇒ IF1(false, x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220)))≥IF2(le(s(s(x214)), size(edge(x112, x113, edge(x222, x221, x220)))), x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220))))






To summarize, we get the following constraints P≥ for the following pairs.
  • IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
    • (IF2(true, x0, x1, x2, edge(x3, x4, edge(x10, x11, x12)), x6)≥IF2(true, x0, x1, x2, edge(x10, x11, x12), x6))
    • (IF2(true, x14, x15, x16, edge(x17, x18, edge(x24, x25, x26)), x20)≥IF2(true, x14, x15, x16, edge(x24, x25, x26), x20))

  • IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)
    • (IF2(true, x56, x57, x58, edge(x59, x60, x61), x62)≥REACH(x60, x57, s(x58), x62, x62))

  • REACH(z4, z1, s(z2), z6, z6) → IF1(eq(z4, z1), z4, z1, s(z2), z6, z6)
    • (REACH(0, s(x124), s(x88), x89, x89)≥IF1(eq(0, s(x124)), 0, s(x124), s(x88), x89, x89))
    • (REACH(s(x125), 0, s(x88), x89, x89)≥IF1(eq(s(x125), 0), s(x125), 0, s(x88), x89, x89))
    • (REACH(x127, x126, s(x88), x89, x89)≥IF1(eq(x127, x126), x127, x126, s(x88), x89, x89) ⇒ REACH(s(x127), s(x126), s(x88), x89, x89)≥IF1(eq(s(x127), s(x126)), s(x127), s(x126), s(x88), x89, x89))

  • IF1(false, z0, z1, s(z2), z3, z3) → IF2(le(s(z2), size(z3)), z0, z1, s(z2), z3, z3)
    • (IF1(false, x94, x95, s(0), edge(x101, x102, x143), edge(x101, x102, x143))≥IF2(le(s(0), size(edge(x101, x102, x143))), x94, x95, s(0), edge(x101, x102, x143), edge(x101, x102, x143)))
    • (IF1(false, x94, x95, s(x162), edge(x101, x102, x168), edge(x101, x102, x168))≥IF2(le(s(x162), size(edge(x101, x102, x168))), x94, x95, s(x162), edge(x101, x102, x168), edge(x101, x102, x168)) ⇒ IF1(false, x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168)))≥IF2(le(s(s(x162)), size(edge(x101, x102, edge(x170, x169, x168)))), x94, x95, s(s(x162)), edge(x101, x102, edge(x170, x169, x168)), edge(x101, x102, edge(x170, x169, x168))))
    • (IF1(false, x105, x106, s(0), edge(x112, x113, x195), edge(x112, x113, x195))≥IF2(le(s(0), size(edge(x112, x113, x195))), x105, x106, s(0), edge(x112, x113, x195), edge(x112, x113, x195)))
    • (IF1(false, x105, x106, s(x214), edge(x112, x113, x220), edge(x112, x113, x220))≥IF2(le(s(x214), size(edge(x112, x113, x220))), x105, x106, s(x214), edge(x112, x113, x220), edge(x112, x113, x220)) ⇒ IF1(false, x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220)))≥IF2(le(s(s(x214)), size(edge(x112, x113, edge(x222, x221, x220)))), x105, x106, s(s(x214)), edge(x112, x113, edge(x222, x221, x220)), edge(x112, x113, edge(x222, x221, x220))))




The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [NONINF]:

POL(0) = 0   
POL(IF1(x1, x2, x3, x4, x5, x6)) = 1 - x4 + x5   
POL(IF2(x1, x2, x3, x4, x5, x6)) = -x4 + x6   
POL(REACH(x1, x2, x3, x4, x5)) = 1 - x3 + x5   
POL(c) = -1   
POL(edge(x1, x2, x3)) = 1 + x3   
POL(empty) = 0   
POL(eq(x1, x2)) = 1 + x1   
POL(false) = 0   
POL(le(x1, x2)) = x2   
POL(s(x1)) = 1 + x1   
POL(size(x1)) = 1 + x1   
POL(true) = 1   

The following pairs are in P>:

IF1(false, z0, z1, s(z2), z3, z3) → IF2(le(s(z2), size(z3)), z0, z1, s(z2), z3, z3)
The following pairs are in Pbound:

IF1(false, z0, z1, s(z2), z3, z3) → IF2(le(s(z2), size(z3)), z0, z1, s(z2), z3, z3)
There are no usable rules

(42) Obligation:

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

IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
IF2(true, x, y, c, edge(u, v, i), j) → REACH(v, y, s(c), j, j)
REACH(z4, z1, s(z2), z6, z6) → IF1(eq(z4, z1), z4, z1, s(z2), z6, z6)

The TRS R consists of the following rules:

size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), s(y)) → le(x, y)
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))
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

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

(43) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(44) Obligation:

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

IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)

The TRS R consists of the following rules:

size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), s(y)) → le(x, y)
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))
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

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

(45) 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.

(46) Obligation:

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

IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)

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))
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

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

(47) 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))
size(empty)
size(edge(x0, x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

(48) Obligation:

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

IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)

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

(49) 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:

  • IF2(true, x, y, c, edge(u, v, i), j) → IF2(true, x, y, c, i, j)
    The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 >= 6

(50) YES