Termination w.r.t. Q of the following Term Rewriting System could be disproven:
Q restricted rewrite system:
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(IL) → ACTIVATE(IL)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
UTAKE1(tt) → NIL
ISNATLIST(n__take(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → ACTIVATE(L)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ACTIVATE(n__0) → 01
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ULENGTH(tt, L) → S(length(activate(L)))
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNAT(N)
ACTIVATE(n__zeros) → ZEROS
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
ACTIVATE(n__nil) → NIL
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → CONS(activate(N), n__take(activate(M), activate(IL)))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → AND(isNat(activate(N)), isNatList(activate(L)))
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(activate(IL)))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(activate(IL))))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
TAKE(0, IL) → UTAKE1(isNatIList(IL))
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(activate(L)))
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ZEROS → CONS(0, n__zeros)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ZEROS → 01
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(IL) → ACTIVATE(IL)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
UTAKE1(tt) → NIL
ISNATLIST(n__take(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → ACTIVATE(L)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ACTIVATE(n__0) → 01
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ULENGTH(tt, L) → S(length(activate(L)))
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNAT(N)
ACTIVATE(n__zeros) → ZEROS
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
ACTIVATE(n__nil) → NIL
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → CONS(activate(N), n__take(activate(M), activate(IL)))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → AND(isNat(activate(N)), isNatList(activate(L)))
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(activate(IL)))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(activate(IL))))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
TAKE(0, IL) → UTAKE1(isNatIList(IL))
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(activate(L)))
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ZEROS → CONS(0, n__zeros)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ZEROS → 01
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 17 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(IL) → ACTIVATE(IL)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → ACTIVATE(L)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNAT(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNAT(n__length(L)) → ACTIVATE(L)
The remaining pairs can at least be oriented weakly.
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(IL) → ACTIVATE(IL)
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → ACTIVATE(L)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNAT(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVATE(x1)) = x1
POL(ISNAT(x1)) = x1
POL(ISNATILIST(x1)) = x1
POL(ISNATLIST(x1)) = x1
POL(LENGTH(x1)) = x1
POL(TAKE(x1, x2)) = x1 + x2
POL(ULENGTH(x1, x2)) = x2
POL(UTAKE2(x1, x2, x3, x4)) = x2 + x3 + x4
POL(activate(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1 + x1
POL(n__0) = 0
POL(n__cons(x1, x2)) = x1 + x2
POL(n__length(x1)) = 1 + x1
POL(n__nil) = 0
POL(n__s(x1)) = x1
POL(n__take(x1, x2)) = x1 + x2
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = 1 + x2
POL(uTake1(x1)) = 0
POL(uTake2(x1, x2, x3, x4)) = x2 + x3 + x4
POL(zeros) = 0
The following usable rules [17] were oriented:
and(tt, T) → T
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
activate(n__0) → 0
take(X1, X2) → n__take(X1, X2)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
zeros → n__zeros
length(X) → n__length(X)
s(X) → n__s(X)
0 → n__0
activate(X) → X
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__zeros) → zeros
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATILIST(IL) → ISNATLIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNAT(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(IL) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
The remaining pairs can at least be oriented weakly.
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ISNATILIST(IL) → ISNATLIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVATE(x1)) = x1
POL(ISNAT(x1)) = 1 + x1
POL(ISNATILIST(x1)) = 1 + x1
POL(ISNATLIST(x1)) = 1 + x1
POL(TAKE(x1, x2)) = 1 + x1 + x2
POL(UTAKE2(x1, x2, x3, x4)) = x2 + x3 + x4
POL(activate(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(n__0) = 0
POL(n__cons(x1, x2)) = x1 + x2
POL(n__length(x1)) = 0
POL(n__nil) = 0
POL(n__s(x1)) = x1
POL(n__take(x1, x2)) = 1 + x1 + x2
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = 0
POL(uTake1(x1)) = 0
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(zeros) = 0
The following usable rules [17] were oriented:
and(tt, T) → T
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
activate(n__0) → 0
take(X1, X2) → n__take(X1, X2)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
zeros → n__zeros
length(X) → n__length(X)
s(X) → n__s(X)
0 → n__0
activate(X) → X
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__zeros) → zeros
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNATILIST(IL) → ISNATLIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 11 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNAT(n__s(N)) → ISNAT(activate(N))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNAT(n__s(N)) → ISNAT(activate(N))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( n__length(x1) ) = | | + | | · | x1 |
| M( n__cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uTake2(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( n__take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uLength(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
and(tt, T) → T
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
activate(n__0) → 0
take(X1, X2) → n__take(X1, X2)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
zeros → n__zeros
length(X) → n__length(X)
s(X) → n__s(X)
0 → n__0
activate(X) → X
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__zeros) → zeros
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L)) at position [0] we obtained the following new rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros)
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(x0))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros)
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(x0))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros) at position [0] we obtained the following new rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__zeros)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__zeros)
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(x0))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(x0))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(x0)) at position [0] we obtained the following new rules:
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(n__s(x0))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(n__s(x0))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1)) at position [0] we obtained the following new rules:
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0) at position [0] we obtained the following new rules:
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(n__0)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(n__0)
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil) at position [0] we obtained the following new rules:
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(n__nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(n__nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
The remaining pairs can at least be oriented weakly.
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ISNATLIST(x1)) = x1
POL(activate(x1)) = 1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(n__0) = 0
POL(n__cons(x1, x2)) = 1 + x2
POL(n__length(x1)) = 1
POL(n__nil) = 0
POL(n__s(x1)) = 0
POL(n__take(x1, x2)) = 0
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 1
POL(tt) = 0
POL(uLength(x1, x2)) = 1
POL(uTake1(x1)) = 0
POL(uTake2(x1, x2, x3, x4)) = 1
POL(zeros) = 0
The following usable rules [17] were oriented:
and(tt, T) → T
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
take(X1, X2) → n__take(X1, X2)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
length(X) → n__length(X)
s(X) → n__s(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
The TRS R consists of the following rules:
0 → n__0
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [15] to enlarge Q to all left-hand sides of R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
The TRS R consists of the following rules:
0 → n__0
The set Q consists of the following terms:
0
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros)) at position [0,0] we obtained the following new rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:
0 → n__0
The set Q consists of the following terms:
0
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
R is empty.
The set Q consists of the following terms:
0
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros)) we obtained the following new rules:
ISNATLIST(n__cons(n__0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(n__0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
ISNATLIST(n__cons(n__0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:none
s = ISNATLIST(n__cons(n__0, n__zeros)) evaluates to t =ISNATLIST(n__cons(n__0, n__zeros))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from ISNATLIST(n__cons(n__0, n__zeros)) to ISNATLIST(n__cons(n__0, n__zeros)).
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.