active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
ACTIVE1(from1(X)) -> FROM1(s1(X))
PROPER1(from1(X)) -> FROM1(proper1(X))
TOP1(mark1(X)) -> PROPER1(X)
PROPER1(after2(X1, X2)) -> PROPER1(X2)
PROPER1(after2(X1, X2)) -> PROPER1(X1)
ACTIVE1(from1(X)) -> FROM1(active1(X))
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X2)
S1(mark1(X)) -> S1(X)
AFTER2(X1, mark1(X2)) -> AFTER2(X1, X2)
ACTIVE1(after2(s1(N), cons2(X, XS))) -> AFTER2(N, XS)
PROPER1(after2(X1, X2)) -> AFTER2(proper1(X1), proper1(X2))
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(s1(X)) -> S1(proper1(X))
AFTER2(ok1(X1), ok1(X2)) -> AFTER2(X1, X2)
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(after2(X1, X2)) -> AFTER2(X1, active1(X2))
ACTIVE1(after2(X1, X2)) -> AFTER2(active1(X1), X2)
AFTER2(mark1(X1), X2) -> AFTER2(X1, X2)
TOP1(ok1(X)) -> ACTIVE1(X)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> S1(X)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
PROPER1(s1(X)) -> PROPER1(X)
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
FROM1(ok1(X)) -> FROM1(X)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
ACTIVE1(from1(X)) -> FROM1(s1(X))
PROPER1(from1(X)) -> FROM1(proper1(X))
TOP1(mark1(X)) -> PROPER1(X)
PROPER1(after2(X1, X2)) -> PROPER1(X2)
PROPER1(after2(X1, X2)) -> PROPER1(X1)
ACTIVE1(from1(X)) -> FROM1(active1(X))
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X2)
S1(mark1(X)) -> S1(X)
AFTER2(X1, mark1(X2)) -> AFTER2(X1, X2)
ACTIVE1(after2(s1(N), cons2(X, XS))) -> AFTER2(N, XS)
PROPER1(after2(X1, X2)) -> AFTER2(proper1(X1), proper1(X2))
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(s1(X)) -> S1(proper1(X))
AFTER2(ok1(X1), ok1(X2)) -> AFTER2(X1, X2)
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(after2(X1, X2)) -> AFTER2(X1, active1(X2))
ACTIVE1(after2(X1, X2)) -> AFTER2(active1(X1), X2)
AFTER2(mark1(X1), X2) -> AFTER2(X1, X2)
TOP1(ok1(X)) -> ACTIVE1(X)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> S1(X)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
PROPER1(s1(X)) -> PROPER1(X)
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
FROM1(ok1(X)) -> FROM1(X)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
AFTER2(ok1(X1), ok1(X2)) -> AFTER2(X1, X2)
AFTER2(mark1(X1), X2) -> AFTER2(X1, X2)
AFTER2(X1, mark1(X2)) -> AFTER2(X1, X2)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
AFTER2(ok1(X1), ok1(X2)) -> AFTER2(X1, X2)
AFTER2(X1, mark1(X2)) -> AFTER2(X1, X2)
Used ordering: Polynomial interpretation [21]:
AFTER2(mark1(X1), X2) -> AFTER2(X1, X2)
POL(AFTER2(x1, x2)) = x2
POL(mark1(x1)) = 1 + x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
AFTER2(mark1(X1), X2) -> AFTER2(X1, X2)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
AFTER2(mark1(X1), X2) -> AFTER2(X1, X2)
POL(AFTER2(x1, x2)) = x1
POL(mark1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
S1(ok1(X)) -> S1(X)
S1(mark1(X)) -> S1(X)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
S1(ok1(X)) -> S1(X)
Used ordering: Polynomial interpretation [21]:
S1(mark1(X)) -> S1(X)
POL(S1(x1)) = x1
POL(mark1(x1)) = x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
S1(mark1(X)) -> S1(X)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
S1(mark1(X)) -> S1(X)
POL(S1(x1)) = x1
POL(mark1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
Used ordering: Polynomial interpretation [21]:
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
POL(CONS2(x1, x2)) = x2
POL(mark1(x1)) = 0
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
POL(CONS2(x1, x2)) = x1
POL(mark1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
FROM1(mark1(X)) -> FROM1(X)
FROM1(ok1(X)) -> FROM1(X)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FROM1(mark1(X)) -> FROM1(X)
Used ordering: Polynomial interpretation [21]:
FROM1(ok1(X)) -> FROM1(X)
POL(FROM1(x1)) = x1
POL(mark1(x1)) = 1 + x1
POL(ok1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
FROM1(ok1(X)) -> FROM1(X)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FROM1(ok1(X)) -> FROM1(X)
POL(FROM1(x1)) = x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
PROPER1(after2(X1, X2)) -> PROPER1(X2)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(after2(X1, X2)) -> PROPER1(X1)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial interpretation [21]:
PROPER1(after2(X1, X2)) -> PROPER1(X2)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(after2(X1, X2)) -> PROPER1(X1)
POL(PROPER1(x1)) = x1
POL(after2(x1, x2)) = x1 + x2
POL(cons2(x1, x2)) = 1 + x1 + x2
POL(from1(x1)) = x1
POL(s1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
PROPER1(after2(X1, X2)) -> PROPER1(X2)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(after2(X1, X2)) -> PROPER1(X1)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER1(after2(X1, X2)) -> PROPER1(X2)
PROPER1(after2(X1, X2)) -> PROPER1(X1)
Used ordering: Polynomial interpretation [21]:
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
POL(PROPER1(x1)) = x1
POL(after2(x1, x2)) = 1 + x1 + x2
POL(from1(x1)) = x1
POL(s1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER1(s1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:
PROPER1(from1(X)) -> PROPER1(X)
POL(PROPER1(x1)) = x1
POL(from1(x1)) = x1
POL(s1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
PROPER1(from1(X)) -> PROPER1(X)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER1(from1(X)) -> PROPER1(X)
POL(PROPER1(x1)) = x1
POL(from1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X1)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
Used ordering: Polynomial interpretation [21]:
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X1)
POL(ACTIVE1(x1)) = x1
POL(after2(x1, x2)) = x1 + x2
POL(cons2(x1, x2)) = 1 + x1
POL(from1(x1)) = x1
POL(s1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X1)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVE1(from1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X1)
POL(ACTIVE1(x1)) = x1
POL(after2(x1, x2)) = x1 + x2
POL(from1(x1)) = 1 + x1
POL(s1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X1)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(after2(X1, X2)) -> ACTIVE1(X1)
Used ordering: Polynomial interpretation [21]:
ACTIVE1(s1(X)) -> ACTIVE1(X)
POL(ACTIVE1(x1)) = x1
POL(after2(x1, x2)) = 1 + x1 + x2
POL(s1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
ACTIVE1(s1(X)) -> ACTIVE1(X)
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVE1(s1(X)) -> ACTIVE1(X)
POL(ACTIVE1(x1)) = x1
POL(s1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
TOP1(ok1(X)) -> TOP1(active1(X))
TOP1(mark1(X)) -> TOP1(proper1(X))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(after2(0, XS)) -> mark1(XS)
active1(after2(s1(N), cons2(X, XS))) -> mark1(after2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(after2(X1, X2)) -> after2(active1(X1), X2)
active1(after2(X1, X2)) -> after2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
after2(mark1(X1), X2) -> mark1(after2(X1, X2))
after2(X1, mark1(X2)) -> mark1(after2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(after2(X1, X2)) -> after2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
after2(ok1(X1), ok1(X2)) -> ok1(after2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))