YES Termination proof of ../tpdb/TRS/CSR_Maude/my-nat/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))
U21(tt, M, N) → U22(tt, M, N)
U22(tt, M, N) → plus(x(N, M), N)
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
s: {1}
plus: {1, 2}
U21: {1}
U22: {1}
x: {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))
U21(tt, M, N) → U22(tt, M, N)
U22(tt, M, N) → plus(x(N, M), N)
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
s: {1}
plus: {1, 2}
U21: {1}
U22: {1}
x: {1, 2}
0: empty set

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

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence

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)))
U21(tt, M, N) → U22(tt, a(M), a(N))
U22(tt, M, N) → plus(x(a(N), a(M)), a(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
a(x) → x

Q is empty.

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

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof

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)))
U21(tt, M, N) → U22(tt, a(M), a(N))
U22(tt, M, N) → plus(x(a(N), a(M)), a(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
a(x) → x

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
a(x0)


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

U111(tt, M, N) → U121(tt, a(M), a(N))
X(N, s(M)) → U211(tt, M, N)
U211(tt, M, N) → U221(tt, a(M), a(N))
U121(tt, M, N) → A(M)
U111(tt, M, N) → A(M)
U221(tt, M, N) → A(M)
PLUS(N, s(M)) → U111(tt, M, N)
U221(tt, M, N) → X(a(N), a(M))
U111(tt, M, N) → A(N)
U221(tt, M, N) → A(N)
U121(tt, M, N) → PLUS(a(N), a(M))
U211(tt, M, N) → A(N)
U221(tt, M, N) → PLUS(x(a(N), a(M)), a(N))
U211(tt, M, N) → A(M)
U121(tt, M, N) → A(N)

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)))
U21(tt, M, N) → U22(tt, a(M), a(N))
U22(tt, M, N) → plus(x(a(N), a(M)), a(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
a(x) → x

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
a(x0)

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

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof

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

U111(tt, M, N) → U121(tt, a(M), a(N))
X(N, s(M)) → U211(tt, M, N)
U211(tt, M, N) → U221(tt, a(M), a(N))
U121(tt, M, N) → A(M)
U111(tt, M, N) → A(M)
U221(tt, M, N) → A(M)
PLUS(N, s(M)) → U111(tt, M, N)
U221(tt, M, N) → X(a(N), a(M))
U111(tt, M, N) → A(N)
U221(tt, M, N) → A(N)
U121(tt, M, N) → PLUS(a(N), a(M))
U211(tt, M, N) → A(N)
U221(tt, M, N) → PLUS(x(a(N), a(M)), a(N))
U211(tt, M, N) → A(M)
U121(tt, M, N) → A(N)

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)))
U21(tt, M, N) → U22(tt, a(M), a(N))
U22(tt, M, N) → plus(x(a(N), a(M)), a(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
a(x) → x

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
a(x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 9 less nodes.

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ UsableRulesProof
                  ↳ QDP

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

U111(tt, M, N) → U121(tt, a(M), a(N))
PLUS(N, s(M)) → U111(tt, M, N)
U121(tt, M, N) → PLUS(a(N), a(M))

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)))
U21(tt, M, N) → U22(tt, a(M), a(N))
U22(tt, M, N) → plus(x(a(N), a(M)), a(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
a(x) → x

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
a(x0)

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ UsableRulesProof
                      ↳ QDP
                        ↳ QReductionProof
                  ↳ QDP

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

U111(tt, M, N) → U121(tt, a(M), a(N))
PLUS(N, s(M)) → U111(tt, M, N)
U121(tt, M, N) → PLUS(a(N), a(M))

The TRS R consists of the following rules:

a(x) → x

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
a(x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ UsableRulesProof
                      ↳ QDP
                        ↳ QReductionProof
                          ↳ QDP
                            ↳ UsableRulesReductionPairsProof
                  ↳ QDP

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

PLUS(N, s(M)) → U111(tt, M, N)
U111(tt, M, N) → U121(tt, a(M), a(N))
U121(tt, M, N) → PLUS(a(N), a(M))

The TRS R consists of the following rules:

a(x) → x

The set Q consists of the following terms:

a(x0)

We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

PLUS(N, s(M)) → U111(tt, M, N)
U121(tt, M, N) → PLUS(a(N), a(M))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(PLUS(x1, x2)) = 2·x1 + 2·x2   
POL(U111(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3   
POL(U121(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + 2·x3   
POL(a(x1)) = x1   
POL(s(x1)) = 2 + x1   
POL(tt) = 1   



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ UsableRulesProof
                      ↳ QDP
                        ↳ QReductionProof
                          ↳ QDP
                            ↳ UsableRulesReductionPairsProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                  ↳ QDP

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

U111(tt, M, N) → U121(tt, a(M), a(N))

The TRS R consists of the following rules:

a(x) → x

The set Q consists of the following terms:

a(x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ UsableRulesProof

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

U221(tt, M, N) → X(a(N), a(M))
U211(tt, M, N) → U221(tt, a(M), a(N))
X(N, s(M)) → U211(tt, M, N)

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)))
U21(tt, M, N) → U22(tt, a(M), a(N))
U22(tt, M, N) → plus(x(a(N), a(M)), a(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
a(x) → x

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
a(x0)

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ UsableRulesProof
                      ↳ QDP
                        ↳ QReductionProof

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

U221(tt, M, N) → X(a(N), a(M))
U211(tt, M, N) → U221(tt, a(M), a(N))
X(N, s(M)) → U211(tt, M, N)

The TRS R consists of the following rules:

a(x) → x

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
a(x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ UsableRulesProof
                      ↳ QDP
                        ↳ QReductionProof
                          ↳ QDP
                            ↳ UsableRulesReductionPairsProof

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

U221(tt, M, N) → X(a(N), a(M))
X(N, s(M)) → U211(tt, M, N)
U211(tt, M, N) → U221(tt, a(M), a(N))

The TRS R consists of the following rules:

a(x) → x

The set Q consists of the following terms:

a(x0)

We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

X(N, s(M)) → U211(tt, M, N)
U211(tt, M, N) → U221(tt, a(M), a(N))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(U211(x1, x2, x3)) = 1 + 2·x1 + x2 + x3   
POL(U221(x1, x2, x3)) = x1 + x2 + x3   
POL(X(x1, x2)) = x1 + x2   
POL(a(x1)) = x1   
POL(s(x1)) = 2 + 2·x1   
POL(tt) = 0   



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ Overlay + Local Confluence
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ UsableRulesProof
                      ↳ QDP
                        ↳ QReductionProof
                          ↳ QDP
                            ↳ UsableRulesReductionPairsProof
                              ↳ QDP
                                ↳ DependencyGraphProof

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

U221(tt, M, N) → X(a(N), a(M))

The TRS R consists of the following rules:

a(x) → x

The set Q consists of the following terms:

a(x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.