Termination w.r.t. Q proof of /tmp/tmp89WCIl/AAECC.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

Q is empty.

↳ QTRS

  ↳ AAECC Innermost

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

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

Q is empty.

We have applied [19,8] to switch to innermost. The TRS R 1 is

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The TRS R 2 is

process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The signature Sigma is {process, if2, if3, if1}

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

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

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

IF1(store, m, false) → EMPTY(fstsplit(m, app(map_f(self, nil), store)))
IF2(store, m, false) → MAP_F(self, nil)
IF1(store, m, false) → MAP_F(self, nil)
IF1(store, m, true) → FSTSPLIT(m, store)
APP(cons(h, t), x) → APP(t, x)
IF1(store, m, true) → EMPTY(fstsplit(m, store))
IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
PROCESS(store, m) → LEQ(m, length(store))
LEQ(s(n), s(m)) → LEQ(n, m)
IF3(store, m, false) → SNDSPLIT(m, app(map_f(self, nil), store))
IF3(store, m, false) → APP(map_f(self, nil), store)
LENGTH(cons(h, t)) → LENGTH(t)
IF1(store, m, false) → FSTSPLIT(m, app(map_f(self, nil), store))
IF2(store, m, false) → SNDSPLIT(m, store)
IF3(store, m, false) → MAP_F(self, nil)
FSTSPLIT(s(n), cons(h, t)) → FSTSPLIT(n, t)
IF3(store, m, false) → PROCESS(sndsplit(m, app(map_f(self, nil), store)), m)
PROCESS(store, m) → LENGTH(store)
MAP_F(pid, cons(h, t)) → MAP_F(pid, t)
IF2(store, m, false) → PROCESS(app(map_f(self, nil), sndsplit(m, store)), m)
IF2(store, m, false) → APP(map_f(self, nil), sndsplit(m, store))
MAP_F(pid, cons(h, t)) → APP(f(pid, h), map_f(pid, t))
IF1(store, m, false) → APP(map_f(self, nil), store)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
SNDSPLIT(s(n), cons(h, t)) → SNDSPLIT(n, t)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

IF1(store, m, false) → EMPTY(fstsplit(m, app(map_f(self, nil), store)))
IF2(store, m, false) → MAP_F(self, nil)
IF1(store, m, false) → MAP_F(self, nil)
IF1(store, m, true) → FSTSPLIT(m, store)
APP(cons(h, t), x) → APP(t, x)
IF1(store, m, true) → EMPTY(fstsplit(m, store))
IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
PROCESS(store, m) → LEQ(m, length(store))
LEQ(s(n), s(m)) → LEQ(n, m)
IF3(store, m, false) → SNDSPLIT(m, app(map_f(self, nil), store))
IF3(store, m, false) → APP(map_f(self, nil), store)
LENGTH(cons(h, t)) → LENGTH(t)
IF1(store, m, false) → FSTSPLIT(m, app(map_f(self, nil), store))
IF2(store, m, false) → SNDSPLIT(m, store)
IF3(store, m, false) → MAP_F(self, nil)
FSTSPLIT(s(n), cons(h, t)) → FSTSPLIT(n, t)
IF3(store, m, false) → PROCESS(sndsplit(m, app(map_f(self, nil), store)), m)
PROCESS(store, m) → LENGTH(store)
MAP_F(pid, cons(h, t)) → MAP_F(pid, t)
IF2(store, m, false) → PROCESS(app(map_f(self, nil), sndsplit(m, store)), m)
IF2(store, m, false) → APP(map_f(self, nil), sndsplit(m, store))
MAP_F(pid, cons(h, t)) → APP(f(pid, h), map_f(pid, t))
IF1(store, m, false) → APP(map_f(self, nil), store)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
SNDSPLIT(s(n), cons(h, t)) → SNDSPLIT(n, t)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 7 SCCs with 15 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

APP(cons(h, t), x) → APP(t, x)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

APP(cons(h, t), x) → APP(t, x)

R is empty.
The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

APP(cons(h, t), x) → APP(t, x)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:

APP(cons(h, t), x) → APP(t, x)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

MAP_F(pid, cons(h, t)) → MAP_F(pid, t)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

MAP_F(pid, cons(h, t)) → MAP_F(pid, t)

R is empty.
The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

MAP_F(pid, cons(h, t)) → MAP_F(pid, t)

From the DPs we obtained the following set of size-change graphs:

MAP_F(pid, cons(h, t)) → MAP_F(pid, t)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LENGTH(cons(h, t)) → LENGTH(t)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LENGTH(cons(h, t)) → LENGTH(t)

R is empty.
The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LENGTH(cons(h, t)) → LENGTH(t)

From the DPs we obtained the following set of size-change graphs:

LENGTH(cons(h, t)) → LENGTH(t)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LEQ(s(n), s(m)) → LEQ(n, m)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LEQ(s(n), s(m)) → LEQ(n, m)

R is empty.
The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LEQ(s(n), s(m)) → LEQ(n, m)

From the DPs we obtained the following set of size-change graphs:

LEQ(s(n), s(m)) → LEQ(n, m)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

SNDSPLIT(s(n), cons(h, t)) → SNDSPLIT(n, t)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

SNDSPLIT(s(n), cons(h, t)) → SNDSPLIT(n, t)

R is empty.
The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

SNDSPLIT(s(n), cons(h, t)) → SNDSPLIT(n, t)

From the DPs we obtained the following set of size-change graphs:

SNDSPLIT(s(n), cons(h, t)) → SNDSPLIT(n, t)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

FSTSPLIT(s(n), cons(h, t)) → FSTSPLIT(n, t)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

FSTSPLIT(s(n), cons(h, t)) → FSTSPLIT(n, t)

R is empty.
The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

FSTSPLIT(s(n), cons(h, t)) → FSTSPLIT(n, t)

From the DPs we obtained the following set of size-change graphs:

FSTSPLIT(s(n), cons(h, t)) → FSTSPLIT(n, t)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
IF2(store, m, false) → PROCESS(app(map_f(self, nil), sndsplit(m, store)), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
IF3(store, m, false) → PROCESS(sndsplit(m, app(map_f(self, nil), store)), m)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
IF2(store, m, false) → PROCESS(app(map_f(self, nil), sndsplit(m, store)), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
IF3(store, m, false) → PROCESS(sndsplit(m, app(map_f(self, nil), store)), m)

The TRS R consists of the following rules:

map_f(pid, nil) → nil
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))
process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

process(x0, x1)
if1(x0, x1, true)
if1(x0, x1, false)
if2(x0, x1, false)
if3(x0, x1, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
IF2(store, m, false) → PROCESS(app(map_f(self, nil), sndsplit(m, store)), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
IF3(store, m, false) → PROCESS(sndsplit(m, app(map_f(self, nil), store)), m)

The TRS R consists of the following rules:

map_f(pid, nil) → nil
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) at position [2,0,1,0] we obtained the following new rules:

IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(nil, store))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF2(store, m, false) → PROCESS(app(map_f(self, nil), sndsplit(m, store)), m)
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(nil, store))))
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
IF3(store, m, false) → PROCESS(sndsplit(m, app(map_f(self, nil), store)), m)

The TRS R consists of the following rules:

map_f(pid, nil) → nil
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF2(store, m, false) → PROCESS(app(map_f(self, nil), sndsplit(m, store)), m) at position [0,0] we obtained the following new rules:

IF2(store, m, false) → PROCESS(app(nil, sndsplit(m, store)), m)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF2(store, m, false) → PROCESS(app(nil, sndsplit(m, store)), m)
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(nil, store))))
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
IF3(store, m, false) → PROCESS(sndsplit(m, app(map_f(self, nil), store)), m)

The TRS R consists of the following rules:

map_f(pid, nil) → nil
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF3(store, m, false) → PROCESS(sndsplit(m, app(map_f(self, nil), store)), m) at position [0,1,0] we obtained the following new rules:

IF3(store, m, false) → PROCESS(sndsplit(m, app(nil, store)), m)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF2(store, m, false) → PROCESS(app(nil, sndsplit(m, store)), m)
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(nil, store))))
IF3(store, m, false) → PROCESS(sndsplit(m, app(nil, store)), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))

The TRS R consists of the following rules:

map_f(pid, nil) → nil
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF2(store, m, false) → PROCESS(app(nil, sndsplit(m, store)), m)
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(nil, store))))
IF3(store, m, false) → PROCESS(sndsplit(m, app(nil, store)), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
app(nil, x) → x
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)
map_f(x0, nil)
map_f(x0, cons(x1, x2))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

map_f(x0, nil)
map_f(x0, cons(x1, x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF2(store, m, false) → PROCESS(app(nil, sndsplit(m, store)), m)
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(nil, store))))
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
IF3(store, m, false) → PROCESS(sndsplit(m, app(nil, store)), m)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
app(nil, x) → x
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF2(store, m, false) → PROCESS(app(nil, sndsplit(m, store)), m) at position [0] we obtained the following new rules:

IF2(store, m, false) → PROCESS(sndsplit(m, store), m)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(nil, store))))
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, app(nil, store)), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
app(nil, x) → x
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, app(nil, store)))) at position [2,0,1] we obtained the following new rules:

IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
IF3(store, m, false) → PROCESS(sndsplit(m, app(nil, store)), m)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
app(nil, x) → x
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF3(store, m, false) → PROCESS(sndsplit(m, app(nil, store)), m) at position [0,1] we obtained the following new rules:

IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
app(nil, x) → x
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))

The TRS R consists of the following rules:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))
app(nil, x0)
app(cons(x0, x1), x2)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

app(nil, x0)
app(cons(x0, x1), x2)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

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

IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store)))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule IF1(store, m, true) → IF2(store, m, empty(fstsplit(m, store))) at position [2] we obtained the following new rules:

IF1(x0, 0, true) → IF2(x0, 0, empty(nil))
IF1(nil, s(x0), true) → IF2(nil, s(x0), empty(nil))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), empty(cons(x1, fstsplit(x0, x2))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

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

IF1(x0, 0, true) → IF2(x0, 0, empty(nil))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), empty(cons(x1, fstsplit(x0, x2))))
IF1(nil, s(x0), true) → IF2(nil, s(x0), empty(nil))
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))

The TRS R consists of the following rules:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

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

IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), empty(cons(x1, fstsplit(x0, x2))))
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), empty(cons(x1, fstsplit(x0, x2)))) at position [2] we obtained the following new rules:

IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

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

IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)
PROCESS(store, m) → IF1(store, m, leq(m, length(store)))

The TRS R consists of the following rules:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule PROCESS(store, m) → IF1(store, m, leq(m, length(store))) at position [2] we obtained the following new rules:

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
PROCESS(nil, y1) → IF1(nil, y1, leq(y1, 0))
PROCESS(y0, 0) → IF1(y0, 0, true)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))
PROCESS(y0, 0) → IF1(y0, 0, true)
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
PROCESS(nil, y1) → IF1(nil, y1, leq(y1, 0))
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store)))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
PROCESS(nil, y1) → IF1(nil, y1, leq(y1, 0))
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule IF1(store, m, false) → IF3(store, m, empty(fstsplit(m, store))) at position [2] we obtained the following new rules:

IF1(nil, s(x0), false) → IF3(nil, s(x0), empty(nil))
IF1(x0, 0, false) → IF3(x0, 0, empty(nil))
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), empty(cons(x1, fstsplit(x0, x2))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF1(nil, s(x0), false) → IF3(nil, s(x0), empty(nil))
IF1(x0, 0, false) → IF3(x0, 0, empty(nil))
PROCESS(nil, y1) → IF1(nil, y1, leq(y1, 0))
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), empty(cons(x1, fstsplit(x0, x2))))
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), empty(cons(x1, fstsplit(x0, x2))))

The TRS R consists of the following rules:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(nil) → true
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), empty(cons(x1, fstsplit(x0, x2))))

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), empty(cons(x1, fstsplit(x0, x2)))) at position [2] we obtained the following new rules:

IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
empty(cons(h, t)) → false
length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QReductionProof

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

fstsplit(0, x0)
fstsplit(s(x0), nil)
fstsplit(s(x0), cons(x1, x2))
empty(nil)
empty(cons(x0, x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QReductionProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF2(store, m, false) → PROCESS(sndsplit(m, store), m)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule IF2(store, m, false) → PROCESS(sndsplit(m, store), m) at position [0] we obtained the following new rules:

IF2(x0, 0, false) → PROCESS(x0, 0)
IF2(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF2(nil, s(x0), false) → PROCESS(nil, s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QReductionProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF2(nil, s(x0), false) → PROCESS(nil, s(x0))
IF2(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF2(x0, 0, false) → PROCESS(x0, 0)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QReductionProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF2(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF3(store, m, false) → PROCESS(sndsplit(m, store), m)

The TRS R consists of the following rules:

length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule IF3(store, m, false) → PROCESS(sndsplit(m, store), m) at position [0] we obtained the following new rules:

IF3(nil, s(x0), false) → PROCESS(nil, s(x0))
IF3(x0, 0, false) → PROCESS(x0, 0)
IF3(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QReductionProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                                                                                                      ↳ QDP

                                                                                                                        ↳ DependencyGraphProof

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

IF3(nil, s(x0), false) → PROCESS(nil, s(x0))
PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF2(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF3(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)
IF3(x0, 0, false) → PROCESS(x0, 0)

The TRS R consists of the following rules:

length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QReductionProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                                                                                                      ↳ QDP

                                                                                                                        ↳ DependencyGraphProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

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

PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1))))
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF2(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF3(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)

The TRS R consists of the following rules:

length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule PROCESS(cons(x0, x1), y1) → IF1(cons(x0, x1), y1, leq(y1, s(length(x1)))) we obtained the following new rules:

PROCESS(cons(x0, x1), s(z2)) → IF1(cons(x0, x1), s(z2), leq(s(z2), s(length(x1))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QReductionProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                                                                                                      ↳ QDP

                                                                                                                        ↳ DependencyGraphProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

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

IF2(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF3(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
PROCESS(cons(x0, x1), s(z2)) → IF1(cons(x0, x1), s(z2), leq(s(z2), s(length(x1))))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)

The TRS R consists of the following rules:

length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule PROCESS(cons(x0, x1), s(z2)) → IF1(cons(x0, x1), s(z2), leq(s(z2), s(length(x1)))) at position [2] we obtained the following new rules:

PROCESS(cons(x0, x1), s(z2)) → IF1(cons(x0, x1), s(z2), leq(z2, length(x1)))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QReductionProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                                                                                                      ↳ QDP

                                                                                                                        ↳ DependencyGraphProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

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

IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF2(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF3(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
PROCESS(cons(x0, x1), s(z2)) → IF1(cons(x0, x1), s(z2), leq(z2, length(x1)))
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)

The TRS R consists of the following rules:

length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

IF1(cons(x1, x2), s(x0), false) → IF3(cons(x1, x2), s(x0), false)
IF1(cons(x1, x2), s(x0), true) → IF2(cons(x1, x2), s(x0), false)

The remaining pairs can at least be oriented weakly.

IF2(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF3(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
PROCESS(cons(x0, x1), s(z2)) → IF1(cons(x0, x1), s(z2), leq(z2, length(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(IF1(x₁, x₂, x₃)) = 1 + x₁
POL(IF2(x₁, x₂, x₃)) = x₁
POL(IF3(x₁, x₂, x₃)) = x₁
POL(PROCESS(x₁, x₂)) = 1 + x₁
POL(cons(x₁, x₂)) = 1 + x₁ + x₂
POL(false) = 0
POL(length(x₁)) = x₁
POL(leq(x₁, x₂)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(sndsplit(x₁, x₂)) = x₂
POL(true) = 0

The following usable rules [17] were oriented:

sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QReductionProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                                                                                                      ↳ QDP

                                                                                                                        ↳ DependencyGraphProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ DependencyGraphProof

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

IF2(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
IF3(cons(x1, x2), s(x0), false) → PROCESS(sndsplit(x0, x2), s(x0))
PROCESS(cons(x0, x1), s(z2)) → IF1(cons(x0, x1), s(z2), leq(z2, length(x1)))

The TRS R consists of the following rules:

length(nil) → 0
length(cons(h, t)) → s(length(t))
leq(0, m) → true
leq(s(n), s(m)) → leq(n, m)
leq(s(n), 0) → false
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)

The set Q consists of the following terms:

sndsplit(0, x0)
sndsplit(s(x0), nil)
sndsplit(s(x0), cons(x1, x2))
leq(0, x0)
leq(s(x0), 0)
leq(s(x0), s(x1))
length(nil)
length(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.