Termination of the following Term Rewriting System could be proven:

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

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

The replacement map contains the following entries:

U101: {1}
tt: empty set
U102: {1}
isNatKind: empty set
U103: {1}
isNat: empty set
U104: {1}
plus: {1, 2}
x: {1, 2}
U11: {1}
U12: {1}
U13: {1}
U14: {1}
U15: {1}
U16: {1}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U33: {1}
U34: {1}
U35: {1}
U36: {1}
U41: {1}
U42: {1}
U51: {1}
U61: {1}
U62: {1}
U71: {1}
U72: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
s: {1}
U91: {1}
U92: {1}
0: empty set


↳ CSR
  ↳ CSDependencyPairsProof

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

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

The replacement map contains the following entries:

U101: {1}
tt: empty set
U102: {1}
isNatKind: empty set
U103: {1}
isNat: empty set
U104: {1}
plus: {1, 2}
x: {1, 2}
U11: {1}
U12: {1}
U13: {1}
U14: {1}
U15: {1}
U16: {1}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U33: {1}
U34: {1}
U35: {1}
U36: {1}
U41: {1}
U42: {1}
U51: {1}
U61: {1}
U62: {1}
U71: {1}
U72: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
s: {1}
U91: {1}
U92: {1}
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 {plus, x, U16, U23, U36, U42, U51, U62, s, U92, PLUS, X, U161, U231, U361, U421, U621, U921, U511} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U1021, U1011, U1031, U1041, U121, U111, U131, U141, U151, U221, U211, U321, U311, U331, U341, U351, U411, U611, U721, U711, U821, U811, U831, U841, U911} 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 DPo are:

U1011(tt, M, N) → U1021(isNatKind(M), M, N)
U1011(tt, M, N) → ISNATKIND(M)
U1021(tt, M, N) → U1031(isNat(N), M, N)
U1021(tt, M, N) → ISNAT(N)
U1031(tt, M, N) → U1041(isNatKind(N), M, N)
U1031(tt, M, N) → ISNATKIND(N)
U1041(tt, M, N) → PLUS(x(N, M), N)
U1041(tt, M, N) → X(N, M)
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, V1, V2) → U321(isNatKind(V1), V1, V2)
U311(tt, V1, V2) → ISNATKIND(V1)
U321(tt, V1, V2) → U331(isNatKind(V2), V1, V2)
U321(tt, V1, V2) → ISNATKIND(V2)
U331(tt, V1, V2) → U341(isNatKind(V2), V1, V2)
U331(tt, V1, V2) → ISNATKIND(V2)
U341(tt, V1, V2) → U351(isNat(V1), V2)
U341(tt, V1, V2) → ISNAT(V1)
U351(tt, V2) → U361(isNat(V2))
U351(tt, V2) → ISNAT(V2)
U411(tt, V2) → U421(isNatKind(V2))
U411(tt, V2) → ISNATKIND(V2)
U611(tt, V2) → U621(isNatKind(V2))
U611(tt, V2) → ISNATKIND(V2)
U711(tt, N) → U721(isNatKind(N), N)
U711(tt, N) → ISNATKIND(N)
U811(tt, M, N) → U821(isNatKind(M), M, N)
U811(tt, M, N) → ISNATKIND(M)
U821(tt, M, N) → U831(isNat(N), M, N)
U821(tt, M, N) → ISNAT(N)
U831(tt, M, N) → U841(isNatKind(N), M, N)
U831(tt, M, N) → ISNATKIND(N)
U841(tt, M, N) → PLUS(N, M)
U911(tt, N) → U921(isNatKind(N))
U911(tt, N) → ISNATKIND(N)
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)
ISNAT(x(V1, V2)) → U311(isNatKind(V1), V1, V2)
ISNAT(x(V1, V2)) → ISNATKIND(V1)
ISNATKIND(plus(V1, V2)) → U411(isNatKind(V1), V2)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → U511(isNatKind(V1))
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → U611(isNatKind(V1), V2)
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)
PLUS(N, 0) → U711(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U811(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U911(isNat(N), N)
X(N, 0) → ISNAT(N)
X(N, s(M)) → U1011(isNat(M), M, N)
X(N, s(M)) → ISNAT(M)

The collapsing dependency pairs are DPc:

U1041(tt, M, N) → N
U1041(tt, M, N) → M
U721(tt, N) → N
U841(tt, M, N) → N
U841(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 :

U1041(tt, M, N) → U(N)
U1041(tt, M, N) → U(M)
U721(tt, N) → U(N)
U841(tt, M, N) → U(N)
U841(tt, M, N) → U(M)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 4 SCCs with 38 less nodes.


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

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U411, U611} 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:

U411(tt, V2) → ISNATKIND(V2)
ISNATKIND(plus(V1, V2)) → U411(isNatKind(V1), V2)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → U611(isNatKind(V1), V2)
U611(tt, V2) → ISNATKIND(V2)
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(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)) → U411(isNatKind(V1), V2)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → U611(isNatKind(V1), V2)
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)
The remaining pairs can at least be oriented weakly.

U411(tt, V2) → ISNATKIND(V2)
U611(tt, V2) → ISNATKIND(V2)
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  x1
U411(x1, x2)  =  x2
U611(x1, x2)  =  x2

Subterm Order


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

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U411, U611} 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:

U411(tt, V2) → ISNATKIND(V2)
U611(tt, V2) → ISNATKIND(V2)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(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
          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U121, U111, U131, U141, U151, U211, U221, U311, U321, U331, U341, U351} 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:

U111(tt, V1, V2) → U121(isNatKind(V1), V1, V2)
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)
ISNAT(s(V1)) → U211(isNatKind(V1), V1)
U211(tt, V1) → U221(isNatKind(V1), V1)
U221(tt, V1) → ISNAT(V1)
ISNAT(x(V1, V2)) → U311(isNatKind(V1), V1, V2)
U311(tt, V1, V2) → U321(isNatKind(V1), V1, V2)
U321(tt, V1, V2) → U331(isNatKind(V2), V1, V2)
U331(tt, V1, V2) → U341(isNatKind(V2), V1, V2)
U341(tt, V1, V2) → U351(isNat(V1), V2)
U351(tt, V2) → ISNAT(V2)
U341(tt, V1, V2) → ISNAT(V1)
U141(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

The following rules are not useable and can be deleted:

U101(tt, x0, x1) → U102(isNatKind(x0), x0, x1)
U102(tt, x0, x1) → U103(isNat(x1), x0, x1)
U103(tt, x0, x1) → U104(isNatKind(x1), x0, x1)
U104(tt, x0, x1) → plus(x(x1, x0), x1)
U71(tt, x0) → U72(isNatKind(x0), x0)
U72(tt, x0) → x0
U81(tt, x0, x1) → U82(isNatKind(x0), x0, x1)
U82(tt, x0, x1) → U83(isNat(x1), x0, x1)
U83(tt, x0, x1) → U84(isNatKind(x1), x0, x1)
U84(tt, x0, x1) → s(plus(x1, x0))
U91(tt, x0) → U92(isNatKind(x0))
U92(tt) → 0
plus(x0, 0) → U71(isNat(x0), x0)
plus(x0, s(x1)) → U81(isNat(x1), x1, x0)
x(x0, 0) → U91(isNat(x0), x0)
x(x0, s(x1)) → U101(isNat(x1), x1, x0)


↳ CSR
  ↳ CSDependencyPairsProof
    ↳ QCSDP
      ↳ QCSDependencyGraphProof
        ↳ AND
          ↳ QCSDP
          ↳ QCSDP
            ↳ QCSUsableRulesProof
              ↳ QCSDP
                ↳ QCSDPReductionPairProof
          ↳ QCSDP
          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, s, U51, x, U62, U42, U23, U36, U16} are replacing on all positions.
For all symbols f in {U41, U61, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U121, U111, U131, U141, U151, U211, U221, U311, U321, U331, U341, U351} 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:

U111(tt, V1, V2) → U121(isNatKind(V1), V1, V2)
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)
ISNAT(s(V1)) → U211(isNatKind(V1), V1)
U211(tt, V1) → U221(isNatKind(V1), V1)
U221(tt, V1) → ISNAT(V1)
ISNAT(x(V1, V2)) → U311(isNatKind(V1), V1, V2)
U311(tt, V1, V2) → U321(isNatKind(V1), V1, V2)
U321(tt, V1, V2) → U331(isNatKind(V2), V1, V2)
U331(tt, V1, V2) → U341(isNatKind(V2), V1, V2)
U341(tt, V1, V2) → U351(isNat(V1), V2)
U351(tt, V2) → ISNAT(V2)
U341(tt, V1, V2) → ISNAT(V1)
U141(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U51(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(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))
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U23(tt) → tt
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt

Q is empty.

Using the order
Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNAT(x1)) = 2·x1   
POL(U11(x1, x2, x3)) = 2 + 2·x2 + x3   
POL(U111(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3   
POL(U12(x1, x2, x3)) = 2 + 2·x1 + 2·x2   
POL(U121(x1, x2, x3)) = 2 + 2·x2 + 2·x3   
POL(U13(x1, x2, x3)) = x1   
POL(U131(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3   
POL(U14(x1, x2, x3)) = x1   
POL(U141(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + 2·x3   
POL(U15(x1, x2)) = 0   
POL(U151(x1, x2)) = 2·x2   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 1 + x1 + 2·x2   
POL(U211(x1, x2)) = 2 + 2·x2   
POL(U22(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(U221(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(U23(x1)) = 1 + x1   
POL(U31(x1, x2, x3)) = 2 + x1 + 2·x3   
POL(U311(x1, x2, x3)) = 2 + 2·x2 + 2·x3   
POL(U32(x1, x2, x3)) = 2 + x1   
POL(U321(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3   
POL(U33(x1, x2, x3)) = 2   
POL(U331(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + 2·x3   
POL(U34(x1, x2, x3)) = x1   
POL(U341(x1, x2, x3)) = 1 + 2·x2 + 2·x3   
POL(U35(x1, x2)) = 0   
POL(U351(x1, x2)) = 1 + 2·x2   
POL(U36(x1)) = 0   
POL(U41(x1, x2)) = 0   
POL(U42(x1)) = 2·x1   
POL(U51(x1)) = 0   
POL(U61(x1, x2)) = x1   
POL(U62(x1)) = 2·x1   
POL(isNat(x1)) = x1   
POL(isNatKind(x1)) = 0   
POL(plus(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(s(x1)) = 2 + 2·x1   
POL(tt) = 0   
POL(x(x1, x2)) = 2 + 2·x1 + 2·x2   

the following usable rules

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(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)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt

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

U131(tt, V1, V2) → U141(isNatKind(V2), V1, V2)
U141(tt, V1, V2) → U151(isNat(V1), V2)
ISNAT(plus(V1, V2)) → U111(isNatKind(V1), V1, V2)
ISNAT(s(V1)) → U211(isNatKind(V1), V1)
U211(tt, V1) → U221(isNatKind(V1), V1)
U221(tt, V1) → ISNAT(V1)
ISNAT(x(V1, V2)) → U311(isNatKind(V1), V1, V2)
U321(tt, V1, V2) → U331(isNatKind(V2), V1, V2)
U351(tt, V2) → ISNAT(V2)
U341(tt, V1, V2) → ISNAT(V1)
U141(tt, V1, V2) → ISNAT(V1)

could be oriented strictly and thus removed.
The pairs

U111(tt, V1, V2) → U121(isNatKind(V1), V1, V2)
U121(tt, V1, V2) → U131(isNatKind(V2), V1, V2)
U151(tt, V2) → ISNAT(V2)
U311(tt, V1, V2) → U321(isNatKind(V1), V1, V2)
U331(tt, V1, V2) → U341(isNatKind(V2), V1, V2)
U341(tt, V1, V2) → U351(isNat(V1), V2)

could only be oriented weakly and must be analyzed further.


↳ CSR
  ↳ CSDependencyPairsProof
    ↳ QCSDP
      ↳ QCSDependencyGraphProof
        ↳ AND
          ↳ QCSDP
          ↳ QCSDP
            ↳ QCSUsableRulesProof
              ↳ QCSDP
                ↳ QCSDPReductionPairProof
                  ↳ QCSDP
                    ↳ QCSDependencyGraphProof
          ↳ QCSDP
          ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, s, U51, x, U62, U42, U23, U36, U16} are replacing on all positions.
For all symbols f in {U41, U61, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U121, U111, U131, U151, U321, U311, U341, U331, U351} 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:

U111(tt, V1, V2) → U121(isNatKind(V1), V1, V2)
U121(tt, V1, V2) → U131(isNatKind(V2), V1, V2)
U151(tt, V2) → ISNAT(V2)
U311(tt, V1, V2) → U321(isNatKind(V1), V1, V2)
U331(tt, V1, V2) → U341(isNatKind(V2), V1, V2)
U341(tt, V1, V2) → U351(isNat(V1), V2)

The TRS R consists of the following rules:

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U51(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(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))
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
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 4 less nodes.


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

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92, PLUS} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U831, U821, U841, U811} we have µ(f) = {1}.
The symbols in {isNatKind, isNat} are not replacing on any position.

The TRS P consists of the following rules:

U821(tt, M, N) → U831(isNat(N), M, N)
U831(tt, M, N) → U841(isNatKind(N), M, N)
U841(tt, M, N) → PLUS(N, M)
PLUS(N, s(M)) → U811(isNat(M), M, N)
U811(tt, M, N) → U821(isNatKind(M), M, N)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(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)) → U811(isNat(M), M, N)
The remaining pairs can at least be oriented weakly.

U821(tt, M, N) → U831(isNat(N), M, N)
U831(tt, M, N) → U841(isNatKind(N), M, N)
U841(tt, M, N) → PLUS(N, M)
U811(tt, M, N) → U821(isNatKind(M), M, N)
Used ordering: Combined order from the following AFS and order.
U831(x1, x2, x3)  =  x2
U821(x1, x2, x3)  =  x2
U841(x1, x2, x3)  =  x2
PLUS(x1, x2)  =  x2
U811(x1, x2, x3)  =  x2

Subterm Order


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

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92, PLUS} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U831, U821, U841, U811} we have µ(f) = {1}.
The symbols in {isNatKind, isNat} are not replacing on any position.

The TRS P consists of the following rules:

U821(tt, M, N) → U831(isNat(N), M, N)
U831(tt, M, N) → U841(isNatKind(N), M, N)
U841(tt, M, N) → PLUS(N, M)
U811(tt, M, N) → U821(isNatKind(M), M, N)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

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


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

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92, X} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U1031, U1021, U1041, U1011} we have µ(f) = {1}.
The symbols in {isNatKind, isNat} are not replacing on any position.

The TRS P consists of the following rules:

U1021(tt, M, N) → U1031(isNat(N), M, N)
U1031(tt, M, N) → U1041(isNatKind(N), M, N)
U1041(tt, M, N) → X(N, M)
X(N, s(M)) → U1011(isNat(M), M, N)
U1011(tt, M, N) → U1021(isNatKind(M), M, N)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(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)) → U1011(isNat(M), M, N)
The remaining pairs can at least be oriented weakly.

U1021(tt, M, N) → U1031(isNat(N), M, N)
U1031(tt, M, N) → U1041(isNatKind(N), M, N)
U1041(tt, M, N) → X(N, M)
U1011(tt, M, N) → U1021(isNatKind(M), M, N)
Used ordering: Combined order from the following AFS and order.
U1031(x1, x2, x3)  =  x2
U1021(x1, x2, x3)  =  x2
U1041(x1, x2, x3)  =  x2
X(x1, x2)  =  x2
U1011(x1, x2, x3)  =  x2

Subterm Order


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

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92, X} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U1031, U1021, U1041, U1011} we have µ(f) = {1}.
The symbols in {isNatKind, isNat} are not replacing on any position.

The TRS P consists of the following rules:

U1021(tt, M, N) → U1031(isNat(N), M, N)
U1031(tt, M, N) → U1041(isNatKind(N), M, N)
U1041(tt, M, N) → X(N, M)
U1011(tt, M, N) → U1021(isNatKind(M), M, N)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
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, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

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