Termination proof of /tmp/tmp8jqEVT/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, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U31: {1}
U32: {1}
U41: {1}
U51: {1}
U52: {1}
s: {1}
plus: {1, 2}
U61: {1}
0: empty set
U71: {1}
U72: {1}
x: {1, 2}

↳ CSR

  ↳ CSDependencyPairsProof

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

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U31: {1}
U32: {1}
U41: {1}
U51: {1}
U52: {1}
s: {1}
plus: {1, 2}
U61: {1}
0: empty set
U71: {1}
U72: {1}
x: {1, 2}

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 {U12, U21, U32, s, plus, U61, x, U12¹, U32¹, PLUS, X, U21¹, U61¹} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U11¹, U31¹, U52¹, U51¹, U72¹, U71¹, U41¹} 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:

U11¹(tt, V2) → U12¹(isNat(V2))
U11¹(tt, V2) → ISNAT(V2)
U31¹(tt, V2) → U32¹(isNat(V2))
U31¹(tt, V2) → ISNAT(V2)
U51¹(tt, M, N) → U52¹(isNat(N), M, N)
U51¹(tt, M, N) → ISNAT(N)
U52¹(tt, M, N) → PLUS(N, M)
U71¹(tt, M, N) → U72¹(isNat(N), M, N)
U71¹(tt, M, N) → ISNAT(N)
U72¹(tt, M, N) → PLUS(x(N, M), N)
U72¹(tt, M, N) → X(N, M)
ISNAT(plus(V1, V2)) → U11¹(isNat(V1), V2)
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → U21¹(isNat(V1))
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31¹(isNat(V1), V2)
ISNAT(x(V1, V2)) → ISNAT(V1)
PLUS(N, 0) → U41¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U61¹(isNat(N))
X(N, 0) → ISNAT(N)
X(N, s(M)) → U71¹(isNat(M), M, N)
X(N, s(M)) → ISNAT(M)

The collapsing dependency pairs are DP_c:

U41¹(tt, N) → N
U52¹(tt, M, N) → N
U52¹(tt, M, N) → M
U72¹(tt, M, N) → N
U72¹(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 :

U41¹(tt, N) → U(N)
U52¹(tt, M, N) → U(N)
U52¹(tt, M, N) → U(M)
U72¹(tt, M, N) → U(N)
U72¹(tt, M, N) → U(M)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 3 SCCs with 17 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPSubtermProof

          ↳ QCSDP

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U21, U32, s, plus, U61, x} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U11¹, U31¹} 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)) → U11¹(isNat(V1), V2)
U11¹(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31¹(isNat(V1), V2)
U31¹(tt, V2) → ISNAT(V2)
ISNAT(x(V1, V2)) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), 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)) → U11¹(isNat(V1), V2)
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31¹(isNat(V1), V2)
ISNAT(x(V1, V2)) → ISNAT(V1)

The remaining pairs can at least be oriented weakly.

U11¹(tt, V2) → ISNAT(V2)
U31¹(tt, V2) → ISNAT(V2)

Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = x2
ISNAT(x1) = x1
U31¹(x1, x2) = x2

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPSubtermProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

          ↳ QCSDP

          ↳ QCSDP

U11¹(tt, V2) → ISNAT(V2)
U31¹(tt, V2) → ISNAT(V2)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(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

            ↳ QCSDPSubtermProof

          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U21, U32, s, plus, U61, x, PLUS} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U52¹, U51¹} we have µ(f) = {1}.
The symbols in {isNat} are not replacing on any position.

The TRS P consists of the following rules:

U51¹(tt, M, N) → U52¹(isNat(N), M, N)
U52¹(tt, M, N) → PLUS(N, M)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(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)) → U51¹(isNat(M), M, N)

The remaining pairs can at least be oriented weakly.

U51¹(tt, M, N) → U52¹(isNat(N), M, N)
U52¹(tt, M, N) → PLUS(N, M)

Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2, x3) = x2
U51¹(x1, x2, x3) = x2
PLUS(x1, x2) = x2

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPSubtermProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

          ↳ QCSDP

U51¹(tt, M, N) → U52¹(isNat(N), M, N)
U52¹(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

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 {U12, U21, U32, s, plus, U61, x, X} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U72¹, U71¹} we have µ(f) = {1}.
The symbols in {isNat} are not replacing on any position.

The TRS P consists of the following rules:

U72¹(tt, M, N) → X(N, M)
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(N), M, N)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

Q is empty.

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

X(N, s(M)) → U71¹(isNat(M), M, N)

The remaining pairs can at least be oriented weakly.

U72¹(tt, M, N) → X(N, M)
U71¹(tt, M, N) → U72¹(isNat(N), M, N)

Used ordering: Combined order from the following AFS and order.
X(x1, x2) = x2
U72¹(x1, x2, x3) = x2
U71¹(x1, x2, x3) = x2

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

            ↳ QCSDPSubtermProof

              ↳ QCSDP

                ↳ QCSDependencyGraphProof

U72¹(tt, M, N) → X(N, M)
U71¹(tt, M, N) → U72¹(isNat(N), M, N)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

Q is empty.

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