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 → g(c)
g(a) → b
f(g(X), b) → f(a, X)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

a → g(c)
g(a) → b
f(g(X), b) → f(a, X)

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:

F(g(X), b) → A
F(g(X), b) → F(a, X)
A → G(c)

The TRS R consists of the following rules:

a → g(c)
g(a) → b
f(g(X), b) → f(a, X)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof

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

F(g(X), b) → A
F(g(X), b) → F(a, X)
A → G(c)

The TRS R consists of the following rules:

a → g(c)
g(a) → b
f(g(X), b) → f(a, X)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof

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

F(g(X), b) → F(a, X)
F(g(X), b) → A
A → G(c)

The TRS R consists of the following rules:

a → g(c)
g(a) → b
f(g(X), b) → f(a, X)

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
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ QDPOrderProof

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

F(g(X), b) → F(a, X)

The TRS R consists of the following rules:

a → g(c)
g(a) → b
f(g(X), b) → f(a, X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


F(g(X), b) → F(a, X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
F(x1, x2)  =  F(x1, x2)
g(x1)  =  x1
b  =  b
a  =  a
c  =  c

Recursive path order with status [2].
Precedence:
F2 > a > c
b > a > c

Status:
b: multiset
a: multiset
c: multiset
F2: multiset

The following usable rules [14] were oriented:

a → g(c)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ QDPOrderProof
                ↳ QDP
                  ↳ PisEmptyProof

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

a → g(c)
g(a) → b
f(g(X), b) → f(a, X)

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.