Termination w.r.t. Q proof of /tmp/tmpv9mWGW/thiemann05.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

Q is empty.

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

IF2(false, u, v) → LEFT(u)
LESS_LEAVES(u, v) → ISLEAF(u)
LESS_LEAVES(u, v) → IF1(isLeaf(u), isLeaf(v), u, v)
IF2(false, u, v) → RIGHT(u)
IF2(false, u, v) → RIGHT(v)
CONCAT(cons(u, v), y) → CONCAT(v, y)
IF2(false, u, v) → LEFT(v)
IF2(false, u, v) → CONCAT(left(u), right(u))
LESS_LEAVES(u, v) → ISLEAF(v)
IF2(false, u, v) → CONCAT(left(v), right(v))
IF1(b, false, u, v) → IF2(b, u, v)
IF2(false, u, v) → LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))

The TRS R consists of the following rules:

isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

IF2(false, u, v) → LEFT(u)
LESS_LEAVES(u, v) → ISLEAF(u)
LESS_LEAVES(u, v) → IF1(isLeaf(u), isLeaf(v), u, v)
IF2(false, u, v) → RIGHT(u)
IF2(false, u, v) → RIGHT(v)
CONCAT(cons(u, v), y) → CONCAT(v, y)
IF2(false, u, v) → LEFT(v)
IF2(false, u, v) → CONCAT(left(u), right(u))
LESS_LEAVES(u, v) → ISLEAF(v)
IF2(false, u, v) → CONCAT(left(v), right(v))
IF1(b, false, u, v) → IF2(b, u, v)
IF2(false, u, v) → LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))

The TRS R consists of the following rules:

isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 8 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

CONCAT(cons(u, v), y) → CONCAT(v, y)

The TRS R consists of the following rules:

isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

Q is empty.
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.

CONCAT(cons(u, v), y) → CONCAT(v, y)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(cons(x₁, x₂)) = 1 + (4)x_2
POL(CONCAT(x₁, x₂)) = (4)x_1

The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

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

LESS_LEAVES(u, v) → IF1(isLeaf(u), isLeaf(v), u, v)
IF1(b, false, u, v) → IF2(b, u, v)
IF2(false, u, v) → LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))

The TRS R consists of the following rules:

isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

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