NO
Termination w.r.t. Q proof of ../tpdb/TRS/TRCSR/Ex4_DLMMU04_FR.trs
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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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:
ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATILIST(IL) → ISNATLIST(activate(IL))
ACTIVATE(n__nil) → NIL
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
UTAKE1(tt) → NIL
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
UTAKE2(tt, M, N, IL) → CONS(activate(N), n__take(activate(M), activate(IL)))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ZEROS → 01
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(n__cons(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ISNAT(N)
ACTIVATE(n__0) → 01
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__cons(N, L)) → AND(isNat(activate(N)), isNatList(activate(L)))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(activate(L)))
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__length(X)) → ACTIVATE(X)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(activate(IL))))
ISNATLIST(n__take(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ULENGTH(tt, L) → S(length(activate(L)))
ISNATILIST(IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ZEROS → CONS(0, n__zeros)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(activate(IL)))
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
TAKE(0, IL) → UTAKE1(isNatIList(IL))
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ULENGTH(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNAT(n__length(L)) → 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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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:
ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATILIST(IL) → ISNATLIST(activate(IL))
ACTIVATE(n__nil) → NIL
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
UTAKE1(tt) → NIL
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
UTAKE2(tt, M, N, IL) → CONS(activate(N), n__take(activate(M), activate(IL)))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ZEROS → 01
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(n__cons(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ISNAT(N)
ACTIVATE(n__0) → 01
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__cons(N, L)) → AND(isNat(activate(N)), isNatList(activate(L)))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(activate(L)))
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__length(X)) → ACTIVATE(X)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(activate(IL))))
ISNATLIST(n__take(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ULENGTH(tt, L) → S(length(activate(L)))
ISNATILIST(IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ZEROS → CONS(0, n__zeros)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(activate(IL)))
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
TAKE(0, IL) → UTAKE1(isNatIList(IL))
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ULENGTH(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNAT(n__length(L)) → 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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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:
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ISNAT(n__s(N)) → ACTIVATE(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__s(X)) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → LENGTH(activate(L))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ULENGTH(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNAT(n__length(L)) → 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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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(activate(X))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ISNATILIST(IL) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNAT(n__length(L)) → ACTIVATE(L)
The remaining pairs can at least be oriented weakly.
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNAT(n__s(N)) → ACTIVATE(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ULENGTH(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVATE(x1)) = x1
POL(ISNAT(x1)) = x1
POL(ISNATILIST(x1)) = 1 + x1
POL(ISNATLIST(x1)) = x1
POL(LENGTH(x1)) = x1
POL(TAKE(x1, x2)) = 1 + 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)) = x1
POL(isNatIList(x1)) = 1 + x1
POL(isNatList(x1)) = 1 + x1
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) = 1
POL(n__s(x1)) = x1
POL(n__take(x1, x2)) = 1 + x1 + x2
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = 1 + x2
POL(uTake1(x1)) = 1
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(zeros) = 0
The following usable rules [17] were oriented:
isNatList(n__nil) → tt
activate(n__length(X)) → length(activate(X))
isNat(n__length(L)) → isNatList(activate(L))
uTake1(tt) → nil
take(X1, X2) → n__take(X1, X2)
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
s(X) → n__s(X)
isNatIList(n__zeros) → tt
activate(n__nil) → nil
activate(n__0) → 0
0 → n__0
uLength(tt, L) → s(length(activate(L)))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
activate(n__s(X)) → s(activate(X))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
nil → n__nil
take(0, IL) → uTake1(isNatIList(IL))
and(tt, T) → T
activate(X) → X
length(X) → n__length(X)
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNat(n__s(N)) → isNat(activate(N))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNat(n__0) → tt
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
isNatIList(IL) → isNatList(activate(IL))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNAT(n__s(N)) → ACTIVATE(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → LENGTH(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ULENGTH(tt, L) → ACTIVATE(L)
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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 5 SCCs with 15 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ 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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL)) at position [0] we obtained the following new rules:
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(0)
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(x1)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(0) at position [0] we obtained the following new rules:
ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(n__0)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(n__0)
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil) at position [0] we obtained the following new rules:
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(n__nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(n__nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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.
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(activate(x0), x1))
The remaining pairs can at least be oriented weakly.
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ISNATILIST(x1)) = x1
POL(activate(x1)) = 1 + x1
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) = 1
POL(n__s(x1)) = 1
POL(n__take(x1, x2)) = 0
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x1)) = 1
POL(take(x1, x2)) = 1
POL(tt) = 0
POL(uLength(x1, x2)) = 1
POL(uTake1(x1)) = 1
POL(uTake2(x1, x2, x3, x4)) = 1
POL(zeros) = 1
The following usable rules [17] were oriented:
activate(n__length(X)) → length(activate(X))
isNatList(n__nil) → tt
isNat(n__length(L)) → isNatList(activate(L))
take(X1, X2) → n__take(X1, X2)
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
s(X) → n__s(X)
isNatIList(n__zeros) → tt
uLength(tt, L) → s(length(activate(L)))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
activate(n__s(X)) → s(activate(X))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
nil → n__nil
take(0, IL) → uTake1(isNatIList(IL))
and(tt, T) → T
length(X) → n__length(X)
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNat(n__s(N)) → isNat(activate(N))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNat(n__0) → tt
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
isNatIList(IL) → isNatList(activate(IL))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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.
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(x1)))
The remaining pairs can at least be oriented weakly.
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( n__length(x1) ) = | | + | | · | x1 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uLength(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uTake2(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( n__cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( n__take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
| M( ISNATILIST(x1) ) = | 0 | + | | · | x1 |
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:
activate(n__length(X)) → length(activate(X))
isNatList(n__nil) → tt
isNat(n__length(L)) → isNatList(activate(L))
take(X1, X2) → n__take(X1, X2)
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
s(X) → n__s(X)
isNatIList(n__zeros) → tt
activate(n__nil) → nil
activate(n__0) → 0
0 → n__0
uLength(tt, L) → s(length(activate(L)))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
activate(n__s(X)) → s(activate(X))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
nil → n__nil
take(0, IL) → uTake1(isNatIList(IL))
and(tt, T) → T
activate(X) → X
length(X) → n__length(X)
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNat(n__s(N)) → isNat(activate(N))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNat(n__0) → tt
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
isNatIList(IL) → isNatList(activate(IL))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
zeros → n__zeros
Used ordering: POLO with Polynomial interpretation [25]:
POL(0) = 0
POL(ISNATILIST(x1)) = 2·x1
POL(cons(x1, x2)) = 1 + 2·x1 + 2·x2
POL(n__0) = 0
POL(n__cons(x1, x2)) = 1 + 2·x1 + 2·x2
POL(n__zeros) = 0
POL(zeros) = 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
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
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
The set Q consists of the following terms:
zeros
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros) at position [0] we obtained the following new rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
The set Q consists of the following terms:
zeros
0
cons(x0, x1)
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
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
The TRS R consists of the following rules:
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
The set Q consists of the following terms:
zeros
0
cons(x0, x1)
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.
zeros
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
The TRS R consists of the following rules:
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
The set Q consists of the following terms:
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
The TRS R consists of the following rules:
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
The set Q consists of the following terms:
0
cons(x0, x1)
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
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(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
cons(x0, x1)
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.
cons(x0, x1)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(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 ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros)) at position [0,0] we obtained the following new rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(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
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(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
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(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 ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros)) we obtained the following new rules:
ISNATILIST(n__cons(n__0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(n__0, n__zeros)) → ISNATILIST(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:
ISNATILIST(n__cons(n__0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:none
s = ISNATILIST(n__cons(n__0, n__zeros)) evaluates to t =ISNATILIST(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 ISNATILIST(n__cons(n__0, n__zeros)) to ISNATILIST(n__cons(n__0, n__zeros)).
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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
↳ 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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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
↳ QDP
↳ 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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(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
↳ QDP
↳ 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(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.