from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)
from: {1}
cons: {1}
s: {1}
head: {1}
2nd: {1}
take: {1, 2}
0: empty set
nil: empty set
sel: {1, 2}
↳ CSR
↳ CSRInnermostProof
from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)
from: {1}
cons: {1}
s: {1}
head: {1}
2nd: {1}
take: {1, 2}
0: empty set
nil: empty set
sel: {1, 2}
The CSR is orthogonal. By [10] we can switch to innermost.
↳ CSR
↳ CSRInnermostProof
↳ CSR
↳ CSDependencyPairsProof
from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)
from: {1}
cons: {1}
s: {1}
head: {1}
2nd: {1}
take: {1, 2}
0: empty set
nil: empty set
sel: {1, 2}
Innermost Strategy.
Using Improved CS-DPs we result in the following initial Q-CSDP problem.
↳ CSR
↳ CSRInnermostProof
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
2ND(cons(X, XS)) → HEAD(XS)
SEL(s(N), cons(X, XS)) → SEL(N, XS)
2ND(cons(X, XS)) → XS
SEL(s(N), cons(X, XS)) → XS
from(s(X))
take(N, XS)
s on positions {1}
from on positions {1}
take on positions {1, 2}
2ND(cons(X, XS)) → U(XS)
SEL(s(N), cons(X, XS)) → U(XS)
U(s(x_0)) → U(x_0)
U(from(x_0)) → U(x_0)
U(take(x_0, x_1)) → U(x_0)
U(take(x_0, x_1)) → U(x_1)
U(from(s(X))) → FROM(s(X))
U(take(N, XS)) → TAKE(N, XS)
from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)
from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
↳ CSR
↳ CSRInnermostProof
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPSubtermProof
↳ QCSDP
U(s(x_0)) → U(x_0)
U(from(x_0)) → U(x_0)
U(take(x_0, x_1)) → U(x_0)
U(take(x_0, x_1)) → U(x_1)
from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)
from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
U(s(x_0)) → U(x_0)
U(from(x_0)) → U(x_0)
U(take(x_0, x_1)) → U(x_0)
U(take(x_0, x_1)) → U(x_1)
↳ CSR
↳ CSRInnermostProof
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPSubtermProof
↳ QCSDP
↳ PIsEmptyProof
↳ QCSDP
from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)
from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
↳ CSR
↳ CSRInnermostProof
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDPSubtermProof
SEL(s(N), cons(X, XS)) → SEL(N, XS)
from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)
from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SEL(s(N), cons(X, XS)) → SEL(N, XS)
↳ CSR
↳ CSRInnermostProof
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDPSubtermProof
↳ QCSDP
↳ PIsEmptyProof
from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)
from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))