YES Termination w.r.t. Q proof of ../tpdb/qualif/ack.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:

a(0, y) → s(y)
a(s(x), 0) → a(x, s(0))
a(s(x), s(y)) → a(x, a(s(x), y))

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

a(0, y) → s(y)
a(s(x), 0) → a(x, s(0))
a(s(x), s(y)) → a(x, a(s(x), y))

Q is empty.

The TRS is overlay and locally confluent. By [19] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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

a(0, y) → s(y)
a(s(x), 0) → a(x, s(0))
a(s(x), s(y)) → a(x, a(s(x), y))

The set Q consists of the following terms:

a(0, x0)
a(s(x0), 0)
a(s(x0), s(x1))


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

A(s(x), s(y)) → A(s(x), y)
A(s(x), s(y)) → A(x, a(s(x), y))
A(s(x), 0) → A(x, s(0))

The TRS R consists of the following rules:

a(0, y) → s(y)
a(s(x), 0) → a(x, s(0))
a(s(x), s(y)) → a(x, a(s(x), y))

The set Q consists of the following terms:

a(0, x0)
a(s(x0), 0)
a(s(x0), s(x1))

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPSizeChangeProof

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

A(s(x), s(y)) → A(s(x), y)
A(s(x), s(y)) → A(x, a(s(x), y))
A(s(x), 0) → A(x, s(0))

The TRS R consists of the following rules:

a(0, y) → s(y)
a(s(x), 0) → a(x, s(0))
a(s(x), s(y)) → a(x, a(s(x), y))

The set Q consists of the following terms:

a(0, x0)
a(s(x0), 0)
a(s(x0), s(x1))

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: