Termination proof of /tmp/tmprmJYiB/MYNAT

Termination of the following Term Rewriting System could be proven:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set

↳ CSR

  ↳ CSDependencyPairsProof

Context-sensitive rewrite system:
The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set

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

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

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

The ordinary context-sensitive dependency pairs DP_o are:

U21¹(tt, M, N) → PLUS(N, M)
ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
PLUS(N, 0) → U11¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U21¹(and(isNat(M), isNat(N)), M, N)
PLUS(N, s(M)) → AND(isNat(M), isNat(N))
PLUS(N, s(M)) → ISNAT(M)

The collapsing dependency pairs are DP_c:

U11¹(tt, N) → N
U21¹(tt, M, N) → N
U21¹(tt, M, N) → M
AND(tt, X) → X

The hidden terms of R are:

isNat(V2)

Every hiding context is built from:none

Hence, the new unhiding pairs DP_u are :

U11¹(tt, N) → U(N)
U21¹(tt, M, N) → U(N)
U21¹(tt, M, N) → U(M)
AND(tt, X) → U(X)
U(isNat(V2)) → ISNAT(V2)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs with 7 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus} are replacing on all positions.
For all symbols f in {U11, U21, and, AND} we have µ(f) = {1}.
The symbols in {isNat, U, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

AND(tt, X) → U(X)
U(isNat(V2)) → ISNAT(V2)
ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

Q is empty.

Using the order
Polynomial interpretation with max and min functions [25]:

POL(0) = 1
POL(AND(x₁, x₂)) = x₁ + x₂
POL(ISNAT(x₁)) = x₁
POL(U(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁, x₂, x₃)) = x₂ + x₃
POL(and(x₁, x₂)) = x₂
POL(isNat(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 1

the following usable rules

isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X

could all be oriented weakly.
Furthermore, the pairs

AND(tt, X) → U(X)

could be oriented strictly and thus removed.
The pairs

U(isNat(V2)) → ISNAT(V2)
ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus} are replacing on all positions.
For all symbols f in {U11, U21, and, AND} we have µ(f) = {1}.
The symbols in {isNat, ISNAT, U} are not replacing on any position.

The TRS P consists of the following rules:

U(isNat(V2)) → ISNAT(V2)
ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 1 SCC with 2 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

                  ↳ QCSDP

                    ↳ QCSDPSubtermProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus} are replacing on all positions.
For all symbols f in {U11, U21, and} we have µ(f) = {1}.
The symbols in {isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

Q is empty.

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
ISNAT(x1) = x1

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

                  ↳ QCSDP

                    ↳ QCSDPSubtermProof

                      ↳ QCSDP

                        ↳ PIsEmptyProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus} are replacing on all positions.
For all symbols f in {U11, U21, and} we have µ(f) = {1}.
The symbols in {isNat} are not replacing on any position.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

Q is empty.

The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPSubtermProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, plus, PLUS} are replacing on all positions.
For all symbols f in {U11, U21, and, U21¹} we have µ(f) = {1}.
The symbols in {isNat} are not replacing on any position.

The TRS P consists of the following rules:

PLUS(N, s(M)) → U21¹(and(isNat(M), isNat(N)), M, N)
U21¹(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

Q is empty.

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

PLUS(N, s(M)) → U21¹(and(isNat(M), isNat(N)), M, N)

The remaining pairs can at least be oriented weakly.

U21¹(tt, M, N) → PLUS(N, M)

Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2, x3) = x2
PLUS(x1, x2) = x2

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPSubtermProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

U21¹(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 0 SCCs.