YES Termination proof of ../tpdb/TRS/CSR_Maude/peanoSimple/MYNAT_nosorts-noand.trs
Termination of the following Term Rewriting System could be proven:

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

U11(tt, M, N) → U12(tt, M, N)
U12(tt, M, N) → s(plus(N, M))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
s: {1}
plus: {1, 2}
0: empty set


↳ CSR
  ↳ Zantema-Transformation

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

U11(tt, M, N) → U12(tt, M, N)
U12(tt, M, N) → s(plus(N, M))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
s: {1}
plus: {1, 2}
0: empty set

We applied the Zantema [34] to transform the context-sensitive TRS to an usual TRS.

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

U11(tt, M, N) → U12(tt, a(M), a(N))
U12(tt, M, N) → s(plus(a(N), a(M)))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
a(x) → x

Q is empty.

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

U11(tt, M, N) → U12(tt, a(M), a(N))
U12(tt, M, N) → s(plus(a(N), a(M)))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
a(x) → x

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

plus(N, 0) → N
Used ordering:
Polynomial interpretation [25]:

POL(0) = 2   
POL(U11(x1, x2, x3)) = x1 + x2 + x3   
POL(U12(x1, x2, x3)) = 2·x1 + x2 + x3   
POL(a(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   




↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof

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

U11(tt, M, N) → U12(tt, a(M), a(N))
U12(tt, M, N) → s(plus(a(N), a(M)))
plus(N, s(M)) → U11(tt, M, N)
a(x) → x

Q is empty.

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

U11(tt, M, N) → U12(tt, a(M), a(N))
U12(tt, M, N) → s(plus(a(N), a(M)))
plus(N, s(M)) → U11(tt, M, N)
a(x) → x

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

plus(N, s(M)) → U11(tt, M, N)
Used ordering:
Polynomial interpretation [25]:

POL(U11(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(U12(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(a(x1)) = x1   
POL(plus(x1, x2)) = 2·x1 + 2·x2   
POL(s(x1)) = 1 + x1   
POL(tt) = 1   




↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof

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

U11(tt, M, N) → U12(tt, a(M), a(N))
U12(tt, M, N) → s(plus(a(N), a(M)))
a(x) → x

Q is empty.

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

U11(tt, M, N) → U12(tt, a(M), a(N))
U12(tt, M, N) → s(plus(a(N), a(M)))
a(x) → x

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

U11(tt, M, N) → U12(tt, a(M), a(N))
U12(tt, M, N) → s(plus(a(N), a(M)))
Used ordering:
Polynomial interpretation [25]:

POL(U11(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3   
POL(U12(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3   
POL(a(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = 2·x1   
POL(tt) = 1   




↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
                ↳ QTRS
                  ↳ RRRPoloQTRSProof

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

a(x) → x

Q is empty.

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

a(x) → x

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

a(x) → x
Used ordering:
Polynomial interpretation [25]:

POL(a(x1)) = 1 + x1   




↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ 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.