Termination of the following Term Rewriting System could be proven:

Context-sensitive 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
  ↳ CSRInnermostProof

Context-sensitive 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

The CSR is orthogonal. By [10] we can switch to innermost.

↳ CSR
  ↳ CSRInnermostProof
    ↳ CSR
      ↳ CSDependencyPairsProof

Context-sensitive 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

Innermost Strategy.

Using Improved CS-DPs we result in the following initial Q-CSDP problem.

↳ CSR
  ↳ CSRInnermostProof
    ↳ CSR
      ↳ CSDependencyPairsProof
        ↳ QCSDP
          ↳ QCSDependencyGraphProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus, x, PLUS, X} are replacing on all positions.
For all symbols f in {U11, U12, U21, U22, U121, U111, U221, U211} we have µ(f) = {1}.
The symbols in {U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DPo are:

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

The collapsing dependency pairs are DPc:

U121(tt, M, N) → N
U121(tt, M, N) → M
U221(tt, M, N) → N
U221(tt, M, N) → M


The hidden terms of R are:
none

Every hiding context is built from:none

Hence, the new unhiding pairs DPu are :

U121(tt, M, N) → U(N)
U121(tt, M, N) → U(M)
U221(tt, M, N) → U(N)
U221(tt, M, N) → U(M)

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 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))


The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs with 5 less nodes.


↳ CSR
  ↳ CSRInnermostProof
    ↳ CSR
      ↳ CSDependencyPairsProof
        ↳ QCSDP
          ↳ QCSDependencyGraphProof
            ↳ AND
              ↳ QCSDP
                ↳ QCSDPSubtermProof
              ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus, x, PLUS} are replacing on all positions.
For all symbols f in {U11, U12, U21, U22, U121, U111} we have µ(f) = {1}.

The TRS P consists of the following rules:

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

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 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))


We use the subterm processor [20].


The following pairs can be oriented strictly and are deleted.


PLUS(N, s(M)) → U111(tt, M, N)
The remaining pairs can at least be oriented weakly.

U121(tt, M, N) → PLUS(N, M)
U111(tt, M, N) → U121(tt, M, N)
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  x2
U121(x1, x2, x3)  =  x2
U111(x1, x2, x3)  =  x2

Subterm Order


↳ CSR
  ↳ CSRInnermostProof
    ↳ CSR
      ↳ CSDependencyPairsProof
        ↳ QCSDP
          ↳ QCSDependencyGraphProof
            ↳ AND
              ↳ QCSDP
                ↳ QCSDPSubtermProof
                  ↳ QCSDP
                    ↳ QCSDependencyGraphProof
              ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus, x, PLUS} are replacing on all positions.
For all symbols f in {U11, U12, U21, U22, U121, U111} we have µ(f) = {1}.

The TRS P consists of the following rules:

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

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 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))


The approximation of the Context-Sensitive Dependency Graph contains 0 SCCs with 2 less nodes.


↳ CSR
  ↳ CSRInnermostProof
    ↳ CSR
      ↳ CSDependencyPairsProof
        ↳ QCSDP
          ↳ QCSDependencyGraphProof
            ↳ AND
              ↳ QCSDP
              ↳ QCSDP
                ↳ QCSDPSubtermProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus, x, X} are replacing on all positions.
For all symbols f in {U11, U12, U21, U22, U221, U211} we have µ(f) = {1}.

The TRS P consists of the following rules:

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

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 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))


We use the subterm processor [20].


The following pairs can be oriented strictly and are deleted.


X(N, s(M)) → U211(tt, M, N)
The remaining pairs can at least be oriented weakly.

U221(tt, M, N) → X(N, M)
U211(tt, M, N) → U221(tt, M, N)
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  x2
U221(x1, x2, x3)  =  x2
U211(x1, x2, x3)  =  x2

Subterm Order


↳ CSR
  ↳ CSRInnermostProof
    ↳ CSR
      ↳ CSDependencyPairsProof
        ↳ QCSDP
          ↳ QCSDependencyGraphProof
            ↳ AND
              ↳ QCSDP
              ↳ QCSDP
                ↳ QCSDPSubtermProof
                  ↳ QCSDP
                    ↳ QCSDependencyGraphProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus, x, X} are replacing on all positions.
For all symbols f in {U11, U12, U21, U22, U221, U211} we have µ(f) = {1}.

The TRS P consists of the following rules:

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

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 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))


The approximation of the Context-Sensitive Dependency Graph contains 0 SCCs with 2 less nodes.