Termination w.r.t. Q proof of TRS/secret06matchbox-gen-10.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:

f₁(b₂(a, z)) -> z
b₂(y, b₂(a, z)) -> b₂(f₁(c₃(y, y, a)), b₂(f₁(z), a))
f₁(f₁(f₁(c₃(z, x, a)))) -> b₂(f₁(x), z)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₁(b₂(a, z)) -> z
b₂(y, b₂(a, z)) -> b₂(f₁(c₃(y, y, a)), b₂(f₁(z), a))
f₁(f₁(f₁(c₃(z, x, a)))) -> b₂(f₁(x), z)

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

B₂(y, b₂(a, z)) -> B₂(f₁(c₃(y, y, a)), b₂(f₁(z), a))
B₂(y, b₂(a, z)) -> F₁(c₃(y, y, a))
F₁(f₁(f₁(c₃(z, x, a)))) -> F₁(x)
B₂(y, b₂(a, z)) -> F₁(z)
B₂(y, b₂(a, z)) -> B₂(f₁(z), a)
F₁(f₁(f₁(c₃(z, x, a)))) -> B₂(f₁(x), z)

The TRS R consists of the following rules:

f₁(b₂(a, z)) -> z
b₂(y, b₂(a, z)) -> b₂(f₁(c₃(y, y, a)), b₂(f₁(z), a))
f₁(f₁(f₁(c₃(z, x, a)))) -> b₂(f₁(x), z)

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:

B₂(y, b₂(a, z)) -> B₂(f₁(c₃(y, y, a)), b₂(f₁(z), a))
B₂(y, b₂(a, z)) -> F₁(c₃(y, y, a))
F₁(f₁(f₁(c₃(z, x, a)))) -> F₁(x)
B₂(y, b₂(a, z)) -> F₁(z)
B₂(y, b₂(a, z)) -> B₂(f₁(z), a)
F₁(f₁(f₁(c₃(z, x, a)))) -> B₂(f₁(x), z)

The TRS R consists of the following rules:

f₁(b₂(a, z)) -> z
b₂(y, b₂(a, z)) -> b₂(f₁(c₃(y, y, a)), b₂(f₁(z), a))
f₁(f₁(f₁(c₃(z, x, a)))) -> b₂(f₁(x), z)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

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

B₂(y, b₂(a, z)) -> B₂(f₁(c₃(y, y, a)), b₂(f₁(z), a))
F₁(f₁(f₁(c₃(z, x, a)))) -> F₁(x)
B₂(y, b₂(a, z)) -> F₁(z)
F₁(f₁(f₁(c₃(z, x, a)))) -> B₂(f₁(x), z)

The TRS R consists of the following rules:

f₁(b₂(a, z)) -> z
b₂(y, b₂(a, z)) -> b₂(f₁(c₃(y, y, a)), b₂(f₁(z), a))
f₁(f₁(f₁(c₃(z, x, a)))) -> b₂(f₁(x), z)

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