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:

f(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(X)

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

f(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(X)

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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

f(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(X)

The set Q consists of the following terms:

f(s(x0), x0)
f(x0, c(x0))
f(x0, x0)


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

F(X, c(X)) → F(s(X), X)
F(s(X), X) → F(X, a(X))

The TRS R consists of the following rules:

f(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(X)

The set Q consists of the following terms:

f(s(x0), x0)
f(x0, c(x0))
f(x0, x0)

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

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

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

F(X, c(X)) → F(s(X), X)
F(s(X), X) → F(X, a(X))

The TRS R consists of the following rules:

f(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(X)

The set Q consists of the following terms:

f(s(x0), x0)
f(x0, c(x0))
f(x0, x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof

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

F(X, c(X)) → F(s(X), X)
F(s(X), X) → F(X, a(X))

The TRS R consists of the following rules:

f(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(X)

The set Q consists of the following terms:

f(s(x0), x0)
f(x0, c(x0))
f(x0, x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 2 less nodes.