YES
0 QTRS
↳1 QTRSToCSRProof (⇔, 0 ms)
↳2 CSR
↳3 CSRInnermostProof (⇔, 0 ms)
↳4 CSR
↳5 CSDependencyPairsProof (⇔, 0 ms)
↳6 QCSDP
↳7 QCSDependencyGraphProof (⇔, 0 ms)
↳8 AND
↳9 QCSDP
↳10 QCSDPSubtermProof (⇔, 0 ms)
↳11 QCSDP
↳12 PIsEmptyProof (⇔, 0 ms)
↳13 YES
↳14 QCSDP
↳15 QCSDPSubtermProof (⇔, 0 ms)
↳16 QCSDP
↳17 PIsEmptyProof (⇔, 0 ms)
↳18 YES
active(from(X)) → mark(cons(X, from(s(X))))
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(sel(0, cons(X, Z))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(sel(0, cons(X, Z))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
from: {1}
cons: {1}
s: {1}
first: {1, 2}
0: empty set
nil: empty set
sel: {1, 2}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
from(X) → cons(X, from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
from: {1}
cons: {1}
s: {1}
first: {1, 2}
0: empty set
nil: empty set
sel: {1, 2}
from(X) → cons(X, from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
from: {1}
cons: {1}
s: {1}
first: {1, 2}
0: empty set
nil: empty set
sel: {1, 2}
Innermost Strategy.
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
SEL(s(X), cons(Y, Z)) → Z
from(s(x0))
first(x0, x1)
s on positions {1}
from on positions {1}
first on positions {1, 2}
SEL(s(X), cons(Y, Z)) → U(Z)
U(s(x_0)) → U(x_0)
U(from(x_0)) → U(x_0)
U(first(x_0, x_1)) → U(x_0)
U(first(x_0, x_1)) → U(x_1)
U(from(s(x0))) → FROM(s(x0))
U(first(x0, x1)) → FIRST(x0, x1)
from(X) → cons(X, from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
from(x0)
first(0, x0)
first(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
U(s(x_0)) → U(x_0)
U(from(x_0)) → U(x_0)
U(first(x_0, x_1)) → U(x_0)
U(first(x_0, x_1)) → U(x_1)
from(X) → cons(X, from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
from(x0)
first(0, x0)
first(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(first(x_0, x_1)) → U(x_0)
U(first(x_0, x_1)) → U(x_1)
from(X) → cons(X, from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
from(x0)
first(0, x0)
first(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
from(X) → cons(X, from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
from(x0)
first(0, x0)
first(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(X), cons(Y, Z)) → SEL(X, Z)
from(X) → cons(X, from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
from(x0)
first(0, x0)
first(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))