Termination proof of /tmp/tmprTxMbv/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, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U31: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: empty set

↳ CSR

  ↳ CSDependencyPairsProof

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

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U31: {1}
U41: {1}
s: {1}
plus: {1, 2}
and: {1}
0: empty set
isNatKind: 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 {U13, U22, s, plus, U13¹, U22¹, PLUS} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U41, and, U12¹, U11¹, U21¹, U41¹, AND, U31¹} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, ISNAT, ISNATKIND, U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

U11¹(tt, V1, V2) → U12¹(isNat(V1), V2)
U11¹(tt, V1, V2) → ISNAT(V1)
U12¹(tt, V2) → U13¹(isNat(V2))
U12¹(tt, V2) → ISNAT(V2)
U21¹(tt, V1) → U22¹(isNat(V1))
U21¹(tt, V1) → ISNAT(V1)
U41¹(tt, M, N) → PLUS(N, M)
ISNAT(plus(V1, V2)) → U11¹(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ISNAT(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNAT(plus(V1, V2)) → ISNATKIND(V1)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
ISNAT(s(V1)) → ISNATKIND(V1)
ISNATKIND(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
PLUS(N, 0) → U31¹(and(isNat(N), isNatKind(N)), N)
PLUS(N, 0) → AND(isNat(N), isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)

The collapsing dependency pairs are DP_c:

U31¹(tt, N) → N
U41¹(tt, M, N) → N
U41¹(tt, M, N) → M
AND(tt, X) → X

The hidden terms of R are:

isNatKind(V2)

Every hiding context is built from:

and on positions {1}

Hence, the new unhiding pairs DP_u are :

U31¹(tt, N) → U(N)
U41¹(tt, M, N) → U(N)
U41¹(tt, M, N) → U(M)
AND(tt, X) → U(X)
U(and(x_0, x_1)) → U(x_0)
U(isNatKind(V2)) → ISNATKIND(V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 3 SCCs with 14 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

          ↳ QCSDP

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, s, plus} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U41, and, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, U, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

U(and(x_0, x_1)) → U(x_0)
U(isNatKind(V2)) → ISNATKIND(V2)
ISNATKIND(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
AND(tt, X) → U(X)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(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(ISNATKIND(x₁)) = x₁
POL(U(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = 1
POL(U12(x₁, x₂)) = 1
POL(U13(x₁)) = 1
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 1
POL(U31(x₁, x₂)) = x₂
POL(U41(x₁, x₂, x₃)) = x₂ + x₃
POL(and(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 1

the following usable rules

and(tt, X) → X
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U31(tt, N) → N
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U41(tt, M, N) → s(plus(N, M))

could all be oriented weakly.
Furthermore, the pairs

AND(tt, X) → U(X)

could be oriented strictly and thus removed.
The pairs

U(and(x_0, x_1)) → U(x_0)
U(isNatKind(V2)) → ISNATKIND(V2)
ISNATKIND(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

          ↳ QCSDP

          ↳ QCSDP

U(and(x_0, x_1)) → U(x_0)
U(isNatKind(V2)) → ISNATKIND(V2)
ISNATKIND(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

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

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

                  ↳ AND

                    ↳ QCSDP

                      ↳ QCSDPSubtermProof

                    ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, s, plus} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U41, and} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

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

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

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

                  ↳ AND

                    ↳ QCSDP

                      ↳ QCSDPSubtermProof

                        ↳ QCSDP

                          ↳ PIsEmptyProof

                    ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, s, plus} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U41, and} we have µ(f) = {1}.
The symbols in {isNat, isNatKind} 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, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(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

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

                  ↳ AND

                    ↳ QCSDP

                    ↳ QCSDP

                      ↳ QCSDPSubtermProof

          ↳ QCSDP

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, s, plus} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U41, and} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, U} are not replacing on any position.

The TRS P consists of the following rules:

U(and(x_0, x_1)) → U(x_0)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

U(and(x_0, x_1)) → U(x_0)

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

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

                  ↳ AND

                    ↳ QCSDP

                    ↳ QCSDP

                      ↳ QCSDPSubtermProof

                        ↳ QCSDP

                          ↳ PIsEmptyProof

          ↳ QCSDP

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, s, plus} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U41, and} we have µ(f) = {1}.
The symbols in {isNat, isNatKind} 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, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(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

            ↳ QCSDPReductionPairProof

            ↳ QCSDPReductionPairProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, s, plus} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U41, and, U12¹, U11¹, U21¹} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U12¹(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → U11¹(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U11¹(tt, V1, V2) → U12¹(isNat(V1), V2)
U11¹(tt, V1, V2) → ISNAT(V1)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
U21¹(tt, V1) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

Using the order
Polynomial Order [21,25] with Interpretation:

POL( plus(x₁, x₂) ) = x₁ + x₂ + 1

POL( U12¹(x₁, x₂) ) = 2x₂ + 1

POL( U31(x₁, x₂) ) = x₂

POL( U41(x₁, ..., x₃) ) = x₂ + x₃ + 2

POL( U11¹(x₁, ..., x₃) ) = 2x₂ + 2x₃ + 1

POL( U11(x₁, ..., x₃) ) = 2x₂ + x₃ + 1

POL( U21¹(x₁, x₂) ) = 2x₂ + 1

POL( U12(x₁, x₂) ) = max{0, -1}

POL( isNatKind(x₁) ) = max{0, -1}

POL( and(x₁, x₂) ) = 2x₂

POL( 0 ) = 0

POL( U21(x₁, x₂) ) = max{0, -1}

POL( ISNAT(x₁) ) = 2x₁ + 1

POL( U22(x₁) ) = max{0, -1}

POL( tt ) = 0

POL( U13(x₁) ) = 0

POL( s(x₁) ) = x₁ + 1

POL( isNat(x₁) ) = max{0, 2x₁ - 1}

the following usable rules

plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U31(tt, N) → N
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U41(tt, M, N) → s(plus(N, M))

could all be oriented weakly.
Furthermore, the pairs

ISNAT(plus(V1, V2)) → U11¹(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)

could be oriented strictly and thus removed.
The pairs

U12¹(tt, V2) → ISNAT(V2)
U11¹(tt, V1, V2) → U12¹(isNat(V1), V2)
U11¹(tt, V1, V2) → ISNAT(V1)
U21¹(tt, V1) → ISNAT(V1)

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

            ↳ QCSDPReductionPairProof

          ↳ QCSDP

U12¹(tt, V2) → ISNAT(V2)
U11¹(tt, V1, V2) → U12¹(isNat(V1), V2)
U11¹(tt, V1, V2) → ISNAT(V1)
U21¹(tt, V1) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

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

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

POL(0) = 1
POL(ISNAT(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = x₃
POL(U11¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U12(x₁, x₂)) = x₂
POL(U12¹(x₁, x₂)) = x₁ + x₂
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₂
POL(U21¹(x₁, x₂)) = x₂
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = x₂
POL(U41(x₁, x₂, x₃)) = x₂ + x₃
POL(and(x₁, x₂)) = x₂
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = 1
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 1

the following usable rules

plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U31(tt, N) → N
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U41(tt, M, N) → s(plus(N, M))

could all be oriented weakly.
Furthermore, the pairs

U12¹(tt, V2) → ISNAT(V2)

could be oriented strictly and thus removed.
The pairs

ISNAT(plus(V1, V2)) → U11¹(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U11¹(tt, V1, V2) → U12¹(isNat(V1), V2)
U11¹(tt, V1, V2) → ISNAT(V1)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
U21¹(tt, V1) → ISNAT(V1)

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, s, plus} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U41, and, U11¹, U12¹, U21¹} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

ISNAT(plus(V1, V2)) → U11¹(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U11¹(tt, V1, V2) → U12¹(isNat(V1), V2)
U11¹(tt, V1, V2) → ISNAT(V1)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
U21¹(tt, V1) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPSubtermProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, s, plus, PLUS} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U41, and, U41¹} we have µ(f) = {1}.
The symbols in {isNat, isNatKind} are not replacing on any position.

The TRS P consists of the following rules:

PLUS(N, s(M)) → U41¹(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U41¹(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(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)) → U41¹(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The remaining pairs can at least be oriented weakly.

U41¹(tt, M, N) → PLUS(N, M)

Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3) = x2
PLUS(x1, x2) = x2

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPSubtermProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

U41¹(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
plus(N, 0) → U31(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

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