zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
U211(tt) → NIL
ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ACTIVATE(n__0) → 01
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
LENGTH(nil) → 01
ACTIVATE(n__zeros) → ZEROS
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
U311(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
U111(tt, L) → ACTIVATE(L)
TAKE(0, IL) → U211(isNatIList(IL))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U111(tt, L) → LENGTH(activate(L))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
U111(tt, L) → S(length(activate(L)))
ZEROS → CONS(0, n__zeros)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U311(tt, IL, M, N) → ACTIVATE(N)
ZEROS → 01
ISNAT(n__length(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
U211(tt) → NIL
ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ACTIVATE(n__0) → 01
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
LENGTH(nil) → 01
ACTIVATE(n__zeros) → ZEROS
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
U311(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
U111(tt, L) → ACTIVATE(L)
TAKE(0, IL) → U211(isNatIList(IL))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U111(tt, L) → LENGTH(activate(L))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
U111(tt, L) → S(length(activate(L)))
ZEROS → CONS(0, n__zeros)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U311(tt, IL, M, N) → ACTIVATE(N)
ZEROS → 01
ISNAT(n__length(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
U111(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
U111(tt, L) → LENGTH(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)
ISNAT(n__length(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNAT(n__length(V1)) → ACTIVATE(V1)
Used ordering: Polynomial interpretation [25,35]:
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
U111(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
U111(tt, L) → LENGTH(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))
The value of delta used in the strict ordering is 1.
POL(U31(x1, x2, x3, x4)) = (4)x_2 + (4)x_3 + (4)x_4
POL(n__isNat(x1)) = (4)x_1
POL(ISNATILIST(x1)) = 4 + (4)x_1
POL(activate(x1)) = x_1
POL(and(x1, x2)) = x_2
POL(n__s(x1)) = x_1
POL(n__nil) = 0
POL(U311(x1, x2, x3, x4)) = 4 + x_2 + (4)x_3 + x_4
POL(take(x1, x2)) = (4)x_1 + (4)x_2
POL(ISNATLIST(x1)) = 4 + x_1
POL(TAKE(x1, x2)) = 4 + (4)x_1 + (4)x_2
POL(tt) = 0
POL(U111(x1, x2)) = 4 + (2)x_2
POL(AND(x1, x2)) = 4 + x_2
POL(isNatList(x1)) = x_1
POL(zeros) = 0
POL(isNatIList(x1)) = (4)x_1
POL(s(x1)) = x_1
POL(isNat(x1)) = (4)x_1
POL(nil) = 0
POL(ACTIVATE(x1)) = 4 + x_1
POL(n__isNatIList(x1)) = (4)x_1
POL(LENGTH(x1)) = 4 + (2)x_1
POL(n__length(x1)) = 1 + (4)x_1
POL(n__zeros) = 0
POL(n__isNatList(x1)) = x_1
POL(n__cons(x1, x2)) = (4)x_1 + x_2
POL(U11(x1, x2)) = 1 + (4)x_2
POL(0) = 0
POL(ISNAT(x1)) = 4 + (4)x_1
POL(n__and(x1, x2)) = x_2
POL(cons(x1, x2)) = (4)x_1 + x_2
POL(n__0) = 0
POL(n__take(x1, x2)) = (4)x_1 + (4)x_2
POL(length(x1)) = 1 + (4)x_1
POL(U21(x1)) = x_1
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
isNatList(n__nil) → tt
length(nil) → 0
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
activate(X) → X
activate(n__nil) → nil
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNatIList(n__zeros) → tt
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
U111(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
U111(tt, L) → LENGTH(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
U311(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
Used ordering: Polynomial interpretation [25,35]:
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
U311(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)
The value of delta used in the strict ordering is 4.
POL(U31(x1, x2, x3, x4)) = 2 + x_2 + (2)x_3 + (2)x_4
POL(n__isNat(x1)) = x_1
POL(ISNATILIST(x1)) = (2)x_1
POL(activate(x1)) = x_1
POL(n__s(x1)) = x_1
POL(and(x1, x2)) = x_2
POL(n__nil) = 0
POL(U311(x1, x2, x3, x4)) = (2)x_2 + (4)x_3 + (4)x_4
POL(take(x1, x2)) = 2 + (2)x_1 + x_2
POL(ISNATLIST(x1)) = (2)x_1
POL(TAKE(x1, x2)) = 4 + (4)x_1 + (2)x_2
POL(tt) = 0
POL(AND(x1, x2)) = (2)x_2
POL(isNatList(x1)) = x_1
POL(zeros) = 0
POL(isNatIList(x1)) = x_1
POL(s(x1)) = x_1
POL(isNat(x1)) = x_1
POL(nil) = 0
POL(ACTIVATE(x1)) = (2)x_1
POL(n__isNatIList(x1)) = x_1
POL(n__length(x1)) = x_1
POL(n__zeros) = 0
POL(n__cons(x1, x2)) = (2)x_1 + x_2
POL(n__isNatList(x1)) = x_1
POL(U11(x1, x2)) = x_2
POL(0) = 0
POL(ISNAT(x1)) = (2)x_1
POL(n__and(x1, x2)) = x_2
POL(cons(x1, x2)) = (2)x_1 + x_2
POL(n__0) = 0
POL(n__take(x1, x2)) = 2 + (2)x_1 + x_2
POL(length(x1)) = x_1
POL(U21(x1)) = 2 + x_1
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
isNatList(n__nil) → tt
length(nil) → 0
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
activate(X) → X
activate(n__nil) → nil
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNatIList(n__zeros) → tt
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U311(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)
U311(tt, IL, M, N) → ACTIVATE(M)
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVATE(n__isNat(X)) → ISNAT(X)
Used ordering: Polynomial interpretation [25,35]:
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
The value of delta used in the strict ordering is 2.
POL(U31(x1, x2, x3, x4)) = (4)x_2 + x_4
POL(n__isNat(x1)) = 2 + x_1
POL(ISNATILIST(x1)) = x_1
POL(activate(x1)) = x_1
POL(and(x1, x2)) = x_2
POL(n__s(x1)) = (4)x_1
POL(n__nil) = 0
POL(take(x1, x2)) = x_2
POL(ISNATLIST(x1)) = x_1
POL(tt) = 0
POL(isNatList(x1)) = x_1
POL(AND(x1, x2)) = x_2
POL(zeros) = 0
POL(isNatIList(x1)) = x_1
POL(s(x1)) = (4)x_1
POL(isNat(x1)) = 2 + x_1
POL(nil) = 0
POL(ACTIVATE(x1)) = x_1
POL(n__isNatIList(x1)) = x_1
POL(n__length(x1)) = x_1
POL(n__zeros) = 0
POL(n__isNatList(x1)) = x_1
POL(n__cons(x1, x2)) = x_1 + (4)x_2
POL(U11(x1, x2)) = (4)x_2
POL(0) = 0
POL(ISNAT(x1)) = x_1
POL(n__and(x1, x2)) = x_2
POL(cons(x1, x2)) = x_1 + (4)x_2
POL(n__0) = 0
POL(n__take(x1, x2)) = x_2
POL(length(x1)) = x_1
POL(U21(x1)) = x_1
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
isNatList(n__nil) → tt
length(nil) → 0
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
activate(X) → X
activate(n__nil) → nil
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNatIList(n__zeros) → tt
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
Used ordering: Polynomial interpretation [25,35]:
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
The value of delta used in the strict ordering is 1.
POL(U31(x1, x2, x3, x4)) = 2 + x_2 + x_3 + (4)x_4
POL(n__isNat(x1)) = x_1
POL(ISNATILIST(x1)) = 1 + x_1
POL(activate(x1)) = x_1
POL(and(x1, x2)) = x_2
POL(n__s(x1)) = x_1
POL(n__nil) = 0
POL(take(x1, x2)) = 2 + x_1 + x_2
POL(ISNATLIST(x1)) = x_1
POL(tt) = 0
POL(isNatList(x1)) = x_1
POL(AND(x1, x2)) = x_2
POL(zeros) = 0
POL(isNatIList(x1)) = 1 + x_1
POL(s(x1)) = x_1
POL(isNat(x1)) = x_1
POL(nil) = 0
POL(ACTIVATE(x1)) = x_1
POL(n__isNatIList(x1)) = 1 + x_1
POL(n__length(x1)) = x_1
POL(n__zeros) = 0
POL(n__isNatList(x1)) = x_1
POL(n__cons(x1, x2)) = (4)x_1 + x_2
POL(U11(x1, x2)) = x_1 + x_2
POL(0) = 0
POL(ISNAT(x1)) = x_1
POL(n__and(x1, x2)) = x_2
POL(cons(x1, x2)) = (4)x_1 + x_2
POL(n__0) = 0
POL(n__take(x1, x2)) = 2 + x_1 + x_2
POL(length(x1)) = x_1
POL(U21(x1)) = 1
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
isNatList(n__nil) → tt
length(nil) → 0
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
activate(X) → X
activate(n__nil) → nil
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNatIList(n__zeros) → tt
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
Used ordering: Polynomial interpretation [25,35]:
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
The value of delta used in the strict ordering is 4.
POL(U31(x1, x2, x3, x4)) = (4)x_2 + (2)x_4
POL(n__isNat(x1)) = 2 + (2)x_1
POL(ISNATILIST(x1)) = 4 + (2)x_1
POL(activate(x1)) = x_1
POL(and(x1, x2)) = x_2
POL(n__s(x1)) = (2)x_1
POL(n__nil) = 0
POL(take(x1, x2)) = x_2
POL(ISNATLIST(x1)) = 4 + x_1
POL(tt) = 0
POL(isNatList(x1)) = 2 + x_1
POL(AND(x1, x2)) = (2)x_2
POL(zeros) = 0
POL(isNatIList(x1)) = 2 + x_1
POL(s(x1)) = (2)x_1
POL(isNat(x1)) = 2 + (2)x_1
POL(nil) = 0
POL(ACTIVATE(x1)) = (2)x_1
POL(n__isNatIList(x1)) = 2 + x_1
POL(n__length(x1)) = x_1
POL(n__zeros) = 0
POL(n__cons(x1, x2)) = (2)x_1 + (4)x_2
POL(n__isNatList(x1)) = 2 + x_1
POL(U11(x1, x2)) = (2)x_2
POL(0) = 0
POL(ISNAT(x1)) = 4 + x_1
POL(n__and(x1, x2)) = x_2
POL(cons(x1, x2)) = (2)x_1 + (4)x_2
POL(n__0) = 0
POL(n__take(x1, x2)) = x_2
POL(length(x1)) = x_1
POL(U21(x1)) = 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
isNatList(n__nil) → tt
length(nil) → 0
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
activate(X) → X
activate(n__nil) → nil
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNatIList(n__zeros) → tt
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
ISNAT(n__s(V1)) → ISNAT(activate(V1))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
Used ordering: Polynomial interpretation [25,35]:
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
The value of delta used in the strict ordering is 4.
POL(U31(x1, x2, x3, x4)) = 3 + (3)x_1 + (3)x_2 + (3)x_3 + (3)x_4
POL(n__isNat(x1)) = 3 + (3)x_1
POL(ISNATILIST(x1)) = 0
POL(activate(x1)) = 0
POL(and(x1, x2)) = 3 + x_1 + (3)x_2
POL(n__nil) = 3
POL(n__s(x1)) = 3 + (4)x_1
POL(take(x1, x2)) = 2 + (2)x_2
POL(ISNATLIST(x1)) = 0
POL(tt) = 0
POL(isNatList(x1)) = 2
POL(AND(x1, x2)) = (4)x_2
POL(zeros) = 3
POL(isNatIList(x1)) = 1
POL(s(x1)) = 2 + (4)x_1
POL(isNat(x1)) = 3 + (3)x_1
POL(nil) = 3
POL(ACTIVATE(x1)) = (4)x_1
POL(n__isNatIList(x1)) = 0
POL(n__length(x1)) = 3 + (3)x_1
POL(n__zeros) = 2
POL(n__cons(x1, x2)) = x_1 + (4)x_2
POL(n__isNatList(x1)) = 0
POL(U11(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(0) = 3
POL(n__and(x1, x2)) = 1 + (3)x_1 + (4)x_2
POL(cons(x1, x2)) = 3 + x_1 + (3)x_2
POL(n__0) = 4
POL(n__take(x1, x2)) = 3 + (3)x_1 + x_2
POL(length(x1)) = 3 + (3)x_1
POL(U21(x1)) = 3 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
U111(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X