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)
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
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)
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
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)
U411(tt, N) → N
U521(tt, M, N) → N
U521(tt, M, N) → M
U721(tt, M, N) → N
U721(tt, M, N) → M
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)
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)
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPSubtermProof
↳ QCSDP
↳ QCSDP
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)
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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
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)
Used ordering: Combined order from the following AFS and order.
U111(tt, V2) → ISNAT(V2)
U311(tt, V2) → ISNAT(V2)
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPSubtermProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ QCSDP
↳ QCSDP
U111(tt, V2) → ISNAT(V2)
U311(tt, V2) → ISNAT(V2)
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)
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDPSubtermProof
↳ QCSDP
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)
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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PLUS(N, s(M)) → U511(isNat(M), M, N)
Used ordering: Combined order from the following AFS and order.
U511(tt, M, N) → U521(isNat(N), M, N)
U521(tt, M, N) → PLUS(N, M)
↳ 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)
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)
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDPSubtermProof
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)
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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
X(N, s(M)) → U711(isNat(M), M, N)
Used ordering: Combined order from the following AFS and order.
U721(tt, M, N) → X(N, M)
U711(tt, M, N) → U721(isNat(N), M, N)
↳ 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)
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)