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(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof

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

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

Q is empty.

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

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

g(c(s(X), Y)) → f(c(X, s(Y)))
Used ordering:
Polynomial interpretation [25]:

POL(c(x1, x2)) = 1 + x1 + x2   
POL(f(x1)) = 1 + 2·x1   
POL(g(x1)) = 2 + 2·x1   
POL(s(x1)) = 2 + x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

f(c(X, s(Y))) → f(c(s(X), Y))

Q is empty.

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

f(c(X, s(Y))) → f(c(s(X), Y))

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

f(c(X, s(Y))) → f(c(s(X), Y))
Used ordering:
Polynomial interpretation [25]:

POL(c(x1, x2)) = x1 + 2·x2   
POL(f(x1)) = 2·x1   
POL(s(x1)) = 1 + x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RisEmptyProof

Q restricted rewrite system:
R is empty.
Q is empty.

The TRS R is empty. Hence, termination is trivially proven.