Termination proof of /tmp/tmpwNc_5v/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(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U13: {1}
U14: {1}
U15: {1}
isNat: empty set
U16: {1}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U41: {1}
U51: {1}
U52: {1}
U61: {1}
U62: {1}
U63: {1}
U64: {1}
s: {1}
plus: {1, 2}
0: empty set

↳ CSR

  ↳ CSDependencyPairsProof

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)

The replacement map contains the following entries:

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {U16, U23, U32, U41, s, plus, U16¹, U23¹, U32¹, PLUS, U41¹} are replacing on all positions.
For all symbols f in {U11, U12, U13, U14, U15, U21, U22, U31, U51, U52, U61, U62, U63, U64, U12¹, U11¹, U13¹, U14¹, U15¹, U22¹, U21¹, U31¹, U52¹, U51¹, U62¹, U61¹, U63¹, U64¹} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNATKIND, ISNAT, U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

U11¹(tt, V1, V2) → U12¹(isNatKind(V1), V1, V2)
U11¹(tt, V1, V2) → ISNATKIND(V1)
U12¹(tt, V1, V2) → U13¹(isNatKind(V2), V1, V2)
U12¹(tt, V1, V2) → ISNATKIND(V2)
U13¹(tt, V1, V2) → U14¹(isNatKind(V2), V1, V2)
U13¹(tt, V1, V2) → ISNATKIND(V2)
U14¹(tt, V1, V2) → U15¹(isNat(V1), V2)
U14¹(tt, V1, V2) → ISNAT(V1)
U15¹(tt, V2) → U16¹(isNat(V2))
U15¹(tt, V2) → ISNAT(V2)
U21¹(tt, V1) → U22¹(isNatKind(V1), V1)
U21¹(tt, V1) → ISNATKIND(V1)
U22¹(tt, V1) → U23¹(isNat(V1))
U22¹(tt, V1) → ISNAT(V1)
U31¹(tt, V2) → U32¹(isNatKind(V2))
U31¹(tt, V2) → ISNATKIND(V2)
U51¹(tt, N) → U52¹(isNatKind(N), N)
U51¹(tt, N) → ISNATKIND(N)
U61¹(tt, M, N) → U62¹(isNatKind(M), M, N)
U61¹(tt, M, N) → ISNATKIND(M)
U62¹(tt, M, N) → U63¹(isNat(N), M, N)
U62¹(tt, M, N) → ISNAT(N)
U63¹(tt, M, N) → U64¹(isNatKind(N), M, N)
U63¹(tt, M, N) → ISNATKIND(N)
U64¹(tt, M, N) → PLUS(N, M)
ISNAT(plus(V1, V2)) → U11¹(isNatKind(V1), V1, V2)
ISNAT(plus(V1, V2)) → ISNATKIND(V1)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
ISNAT(s(V1)) → ISNATKIND(V1)
ISNATKIND(plus(V1, V2)) → U31¹(isNatKind(V1), V2)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → U41¹(isNatKind(V1))
ISNATKIND(s(V1)) → ISNATKIND(V1)
PLUS(N, 0) → U51¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)

The collapsing dependency pairs are DP_c:

U52¹(tt, N) → N
U64¹(tt, M, N) → N
U64¹(tt, M, N) → M

The hidden terms of R are:
none

Every hiding context is built from:none

Hence, the new unhiding pairs DP_u are :

U52¹(tt, N) → U(N)
U64¹(tt, M, N) → U(N)
U64¹(tt, M, N) → U(M)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 3 SCCs with 21 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPSubtermProof

          ↳ QCSDP

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U16, U23, U32, U41, s, plus} are replacing on all positions.
For all symbols f in {U11, U12, U13, U14, U15, U21, U22, U31, U51, U52, U61, U62, U63, U64, U31¹} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

U31¹(tt, V2) → ISNATKIND(V2)
ISNATKIND(plus(V1, V2)) → U31¹(isNatKind(V1), 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(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), 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)) → U31¹(isNatKind(V1), V2)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)

The remaining pairs can at least be oriented weakly.

U31¹(tt, V2) → ISNATKIND(V2)

Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1) = x1
U31¹(x1, x2) = x2

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPSubtermProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

          ↳ QCSDP

          ↳ QCSDP

U31¹(tt, V2) → ISNATKIND(V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)

Q is empty.

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

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSUsableRulesProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U16, U23, U32, U41, s, plus} are replacing on all positions.
For all symbols f in {U11, U12, U13, U14, U15, U21, U22, U31, U51, U52, U61, U62, U63, U64, U13¹, U12¹, U14¹, U15¹, U11¹, U21¹, U22¹} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U12¹(tt, V1, V2) → U13¹(isNatKind(V2), V1, V2)
U13¹(tt, V1, V2) → U14¹(isNatKind(V2), V1, V2)
U14¹(tt, V1, V2) → U15¹(isNat(V1), V2)
U15¹(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → U11¹(isNatKind(V1), V1, V2)
U11¹(tt, V1, V2) → U12¹(isNatKind(V1), V1, V2)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
U21¹(tt, V1) → U22¹(isNatKind(V1), V1)
U22¹(tt, V1) → ISNAT(V1)
U14¹(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)

Q is empty.

The following rules are not useable and can be deleted:

U51(tt, x0) → U52(isNatKind(x0), x0)
U52(tt, x0) → x0
U61(tt, x0, x1) → U62(isNatKind(x0), x0, x1)
U62(tt, x0, x1) → U63(isNat(x1), x0, x1)
U63(tt, x0, x1) → U64(isNatKind(x1), x0, x1)
U64(tt, x0, x1) → s(plus(x1, x0))
plus(x0, 0) → U51(isNat(x0), x0)
plus(x0, s(x1)) → U61(isNat(x1), x1, x0)

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSUsableRulesProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, s, U41, U32, U23, U16} are replacing on all positions.
For all symbols f in {U31, U11, U12, U13, U14, U15, U21, U22, U13¹, U12¹, U14¹, U15¹, U11¹, U21¹, U22¹} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U12¹(tt, V1, V2) → U13¹(isNatKind(V2), V1, V2)
U13¹(tt, V1, V2) → U14¹(isNatKind(V2), V1, V2)
U14¹(tt, V1, V2) → U15¹(isNat(V1), V2)
U15¹(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → U11¹(isNatKind(V1), V1, V2)
U11¹(tt, V1, V2) → U12¹(isNatKind(V1), V1, V2)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
U21¹(tt, V1) → U22¹(isNatKind(V1), V1)
U22¹(tt, V1) → ISNAT(V1)
U14¹(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
U41(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt

Q is empty.

Using the order
Polynomial interpretation [25]:

POL(0) = 2
POL(ISNAT(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = x₁
POL(U11¹(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(U12(x₁, x₂, x₃)) = 2·x₁
POL(U12¹(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃
POL(U13(x₁, x₂, x₃)) = 0
POL(U13¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U14(x₁, x₂, x₃)) = 2·x₁
POL(U14¹(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U15(x₁, x₂)) = x₁
POL(U15¹(x₁, x₂)) = 2·x₁ + x₂
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U21¹(x₁, x₂)) = 2·x₁ + x₂
POL(U22(x₁, x₂)) = 0
POL(U22¹(x₁, x₂)) = x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 2·x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = 2·x₁
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(plus(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(s(x₁)) = x₁
POL(tt) = 0

the following usable rules

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt

could all be oriented weakly.
Since all dependency pairs and these rules are strongly conservative, this is sound.
Furthermore, the pairs

U12¹(tt, V1, V2) → U13¹(isNatKind(V2), V1, V2)

could be oriented strictly and thus removed.
The pairs

U13¹(tt, V1, V2) → U14¹(isNatKind(V2), V1, V2)
U14¹(tt, V1, V2) → U15¹(isNat(V1), V2)
U15¹(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → U11¹(isNatKind(V1), V1, V2)
U11¹(tt, V1, V2) → U12¹(isNatKind(V1), V1, V2)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
U21¹(tt, V1) → U22¹(isNatKind(V1), V1)
U22¹(tt, V1) → ISNAT(V1)
U14¹(tt, V1, V2) → ISNAT(V1)

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSUsableRulesProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, s, U41, U32, U23, U16} are replacing on all positions.
For all symbols f in {U31, U11, U12, U13, U14, U15, U21, U22, U14¹, U13¹, U15¹, U11¹, U12¹, U21¹, U22¹} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U13¹(tt, V1, V2) → U14¹(isNatKind(V2), V1, V2)
U14¹(tt, V1, V2) → U15¹(isNat(V1), V2)
U15¹(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → U11¹(isNatKind(V1), V1, V2)
U11¹(tt, V1, V2) → U12¹(isNatKind(V1), V1, V2)
ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
U21¹(tt, V1) → U22¹(isNatKind(V1), V1)
U22¹(tt, V1) → ISNAT(V1)
U14¹(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
U41(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 1 SCC with 6 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSUsableRulesProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ QCSDP

                        ↳ QCSDPSubtermProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, s, U41, U32, U23, U16} are replacing on all positions.
For all symbols f in {U31, U11, U12, U13, U14, U15, U21, U22, U21¹, U22¹} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)
U21¹(tt, V1) → U22¹(isNatKind(V1), V1)
U22¹(tt, V1) → ISNAT(V1)

The TRS R consists of the following rules:

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
U41(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt

Q is empty.

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

ISNAT(s(V1)) → U21¹(isNatKind(V1), V1)

The remaining pairs can at least be oriented weakly.

U21¹(tt, V1) → U22¹(isNatKind(V1), V1)
U22¹(tt, V1) → ISNAT(V1)

Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2) = x2
ISNAT(x1) = x1
U22¹(x1, x2) = x2

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSUsableRulesProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ QCSDP

                        ↳ QCSDPSubtermProof

                          ↳ QCSDP

                            ↳ QCSDependencyGraphProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, s, U41, U32, U23, U16} are replacing on all positions.
For all symbols f in {U31, U11, U12, U13, U14, U15, U21, U22, U22¹, U21¹} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U21¹(tt, V1) → U22¹(isNatKind(V1), V1)
U22¹(tt, V1) → ISNAT(V1)

The TRS R consists of the following rules:

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
U41(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt

Q is empty.

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

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPSubtermProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {U16, U23, U32, U41, s, plus, PLUS} are replacing on all positions.
For all symbols f in {U11, U12, U13, U14, U15, U21, U22, U31, U51, U52, U61, U62, U63, U64, U63¹, U62¹, U64¹, U61¹} we have µ(f) = {1}.
The symbols in {isNatKind, isNat} are not replacing on any position.

The TRS P consists of the following rules:

U62¹(tt, M, N) → U63¹(isNat(N), M, N)
U63¹(tt, M, N) → U64¹(isNatKind(N), M, N)
U64¹(tt, M, N) → PLUS(N, M)
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U61¹(tt, M, N) → U62¹(isNatKind(M), M, N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), 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)) → U61¹(isNat(M), M, N)

The remaining pairs can at least be oriented weakly.

U62¹(tt, M, N) → U63¹(isNat(N), M, N)
U63¹(tt, M, N) → U64¹(isNatKind(N), M, N)
U64¹(tt, M, N) → PLUS(N, M)
U61¹(tt, M, N) → U62¹(isNatKind(M), M, N)

Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3) = x2
U62¹(x1, x2, x3) = x2
U64¹(x1, x2, x3) = x2
PLUS(x1, x2) = x2
U61¹(x1, x2, x3) = x2

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPSubtermProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

U62¹(tt, M, N) → U63¹(isNat(N), M, N)
U63¹(tt, M, N) → U64¹(isNatKind(N), M, N)
U64¹(tt, M, N) → PLUS(N, M)
U61¹(tt, M, N) → U62¹(isNatKind(M), M, N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 0 SCCs with 4 less nodes.