Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
take(0, IL) → nil
Used ordering:
Polynomial interpretation [25]:
POL(0) = 0
POL(U11(x1, x2)) = 2·x1 + 2·x2
POL(U12(x1, x2)) = x1 + 2·x2
POL(U21(x1, x2, x3, x4)) = 1 + x1 + x2 + 2·x3 + 2·x4
POL(U22(x1, x2, x3, x4)) = 1 + 2·x1 + x2 + 2·x3 + 2·x4
POL(U23(x1, x2, x3, x4)) = 1 + 2·x1 + x2 + 2·x3 + 2·x4
POL(activate(x1)) = x1
POL(cons(x1, x2)) = 2·x1 + x2
POL(length(x1)) = 2·x1
POL(n__take(x1, x2)) = 1 + 2·x1 + x2
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + 2·x1 + x2
POL(tt) = 0
POL(zeros) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
length(nil) → 0
Used ordering:
Polynomial interpretation [25]:
POL(0) = 0
POL(U11(x1, x2)) = 2·x1 + 2·x2
POL(U12(x1, x2)) = 2·x1 + 2·x2
POL(U21(x1, x2, x3, x4)) = 2·x1 + 2·x2 + 2·x3 + x4
POL(U22(x1, x2, x3, x4)) = 2·x1 + 2·x2 + 2·x3 + x4
POL(U23(x1, x2, x3, x4)) = x1 + 2·x2 + 2·x3 + x4
POL(activate(x1)) = x1
POL(cons(x1, x2)) = x1 + 2·x2
POL(length(x1)) = x1
POL(n__take(x1, x2)) = x1 + x2
POL(n__zeros) = 0
POL(nil) = 2
POL(s(x1)) = 2·x1
POL(take(x1, x2)) = x1 + x2
POL(tt) = 0
POL(zeros) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → U11(tt, activate(L))
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
U211(tt, IL, M, N) → ACTIVATE(M)
U121(tt, L) → ACTIVATE(L)
U221(tt, IL, M, N) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ACTIVATE(L)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U221(tt, IL, M, N) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(M)
LENGTH(cons(N, L)) → U111(tt, activate(L))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
U211(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
U211(tt, IL, M, N) → ACTIVATE(IL)
U111(tt, L) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
ACTIVATE(n__zeros) → ZEROS
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → U11(tt, activate(L))
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U211(tt, IL, M, N) → ACTIVATE(M)
U121(tt, L) → ACTIVATE(L)
U221(tt, IL, M, N) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ACTIVATE(L)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U221(tt, IL, M, N) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(M)
LENGTH(cons(N, L)) → U111(tt, activate(L))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
U211(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
U211(tt, IL, M, N) → ACTIVATE(IL)
U111(tt, L) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
ACTIVATE(n__zeros) → ZEROS
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → U11(tt, activate(L))
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
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
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, IL, M, N) → ACTIVATE(M)
U221(tt, IL, M, N) → ACTIVATE(IL)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U221(tt, IL, M, N) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(M)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
U211(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
U211(tt, IL, M, N) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → U11(tt, activate(L))
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
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
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, IL, M, N) → ACTIVATE(M)
U221(tt, IL, M, N) → ACTIVATE(IL)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U221(tt, IL, M, N) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(M)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
U211(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
U211(tt, IL, M, N) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
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.
length(cons(x0, x1))
U12(tt, x0)
length(nil)
U11(tt, x0)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, IL, M, N) → ACTIVATE(M)
U221(tt, IL, M, N) → ACTIVATE(IL)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U221(tt, IL, M, N) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(M)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
U211(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
U211(tt, IL, M, N) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
zeros
U23(tt, x0, x1, x2)
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
U211(tt, IL, M, N) → ACTIVATE(M)
U221(tt, IL, M, N) → ACTIVATE(IL)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
U221(tt, IL, M, N) → ACTIVATE(N)
U211(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U221(tt, IL, M, N) → ACTIVATE(M)
U211(tt, IL, M, N) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
The following rules are removed from R:
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
Used ordering: POLO with Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVATE(x1)) = x1
POL(TAKE(x1, x2)) = x1 + 2·x2
POL(U21(x1, x2, x3, x4)) = 1 + x1 + 2·x2 + 2·x3 + 2·x4
POL(U211(x1, x2, x3, x4)) = 1 + 2·x1 + x2 + x3 + 2·x4
POL(U22(x1, x2, x3, x4)) = 2·x1 + 2·x2 + 2·x3 + 2·x4
POL(U221(x1, x2, x3, x4)) = 1 + 2·x1 + x2 + x3 + x4
POL(U23(x1, x2, x3, x4)) = 2·x1 + 2·x2 + x3 + 2·x4
POL(U231(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
POL(activate(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(n__take(x1, x2)) = x1 + 2·x2
POL(n__zeros) = 0
POL(s(x1)) = 1 + 2·x1
POL(take(x1, x2)) = x1 + 2·x2
POL(tt) = 0
POL(zeros) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U231(tt, IL, M, N) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(M)
U231(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
zeros
U23(tt, x0, x1, x2)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 5 less nodes.
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
zeros
U23(tt, x0, x1, x2)
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
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
R is empty.
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
zeros
U23(tt, x0, x1, x2)
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.
take(x0, x1)
activate(x0)
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
zeros
U23(tt, x0, x1, x2)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
The graph contains the following edges 1 > 1
- ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
The graph contains the following edges 1 > 1
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → U11(tt, activate(L))
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
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
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
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.
length(cons(x0, x1))
U12(tt, x0)
length(nil)
U11(tt, x0)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
zeros
U23(tt, x0, x1, x2)
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
No dependency pairs are removed.
The following rules are removed from R:
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
Used ordering: POLO with Polynomial interpretation [25]:
POL(0) = 0
POL(LENGTH(x1)) = x1
POL(U111(x1, x2)) = 2·x1 + x2
POL(U121(x1, x2)) = 2·x1 + x2
POL(U21(x1, x2, x3, x4)) = 1 + 2·x1 + x2 + 2·x3 + x4
POL(U22(x1, x2, x3, x4)) = 1 + 2·x1 + x2 + 2·x3 + x4
POL(U23(x1, x2, x3, x4)) = 1 + x1 + x2 + 2·x3 + x4
POL(activate(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(n__take(x1, x2)) = 2·x1 + x2
POL(n__zeros) = 0
POL(s(x1)) = 2 + x1
POL(take(x1, x2)) = 2·x1 + x2
POL(tt) = 0
POL(zeros) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
zeros
U23(tt, x0, x1, x2)
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
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
zeros
U23(tt, x0, x1, x2)
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.
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
U23(tt, x0, x1, x2)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U121(tt, L) → LENGTH(activate(L)) at position [0] we obtained the following new rules:
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U121(tt, n__zeros) → LENGTH(zeros)
U121(tt, x0) → LENGTH(x0)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U111(tt, L) → U121(tt, activate(L))
U121(tt, n__zeros) → LENGTH(zeros)
LENGTH(cons(N, L)) → U111(tt, activate(L))
U121(tt, x0) → LENGTH(x0)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U111(tt, L) → U121(tt, activate(L)) at position [1] we obtained the following new rules:
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U111(tt, x0) → U121(tt, x0)
U111(tt, n__zeros) → U121(tt, zeros)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U111(tt, n__zeros) → U121(tt, zeros)
U121(tt, n__zeros) → LENGTH(zeros)
LENGTH(cons(N, L)) → U111(tt, activate(L))
U111(tt, x0) → U121(tt, x0)
U121(tt, x0) → LENGTH(x0)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule LENGTH(cons(N, L)) → U111(tt, activate(L)) at position [1] we obtained the following new rules:
LENGTH(cons(y0, x0)) → U111(tt, x0)
LENGTH(cons(y0, n__zeros)) → U111(tt, zeros)
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
LENGTH(cons(y0, n__zeros)) → U111(tt, zeros)
U111(tt, n__zeros) → U121(tt, zeros)
LENGTH(cons(y0, x0)) → U111(tt, x0)
U121(tt, n__zeros) → LENGTH(zeros)
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
U121(tt, x0) → LENGTH(x0)
U111(tt, x0) → U121(tt, x0)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U121(tt, n__zeros) → LENGTH(zeros) at position [0] we obtained the following new rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, n__zeros) → LENGTH(n__zeros)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U121(tt, n__zeros) → LENGTH(n__zeros)
LENGTH(cons(y0, n__zeros)) → U111(tt, zeros)
U111(tt, n__zeros) → U121(tt, zeros)
LENGTH(cons(y0, x0)) → U111(tt, x0)
U111(tt, x0) → U121(tt, x0)
U121(tt, x0) → LENGTH(x0)
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
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
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
LENGTH(cons(y0, n__zeros)) → U111(tt, zeros)
U111(tt, n__zeros) → U121(tt, zeros)
LENGTH(cons(y0, x0)) → U111(tt, x0)
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
U121(tt, x0) → LENGTH(x0)
U111(tt, x0) → U121(tt, x0)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule LENGTH(cons(y0, n__zeros)) → U111(tt, zeros) at position [1] we obtained the following new rules:
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U111(tt, n__zeros) → U121(tt, zeros)
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
LENGTH(cons(y0, x0)) → U111(tt, x0)
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
U111(tt, x0) → U121(tt, x0)
U121(tt, x0) → LENGTH(x0)
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U111(tt, n__zeros) → U121(tt, zeros) at position [1] we obtained the following new rules:
U111(tt, n__zeros) → U121(tt, n__zeros)
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
LENGTH(cons(y0, x0)) → U111(tt, x0)
U111(tt, n__zeros) → U121(tt, n__zeros)
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
U121(tt, x0) → LENGTH(x0)
U111(tt, x0) → U121(tt, x0)
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule U121(tt, x0) → LENGTH(x0) we obtained the following new rules:
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U121(tt, cons(y_0, n__take(y_1, y_2))) → LENGTH(cons(y_0, n__take(y_1, y_2)))
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U121(tt, cons(y_0, n__take(y_1, y_2))) → LENGTH(cons(y_0, n__take(y_1, y_2)))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
LENGTH(cons(y0, x0)) → U111(tt, x0)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
U111(tt, n__zeros) → U121(tt, n__zeros)
U111(tt, x0) → U121(tt, x0)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule U111(tt, x0) → U121(tt, x0) we obtained the following new rules:
U111(tt, n__take(y_0, y_1)) → U121(tt, n__take(y_0, y_1))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
U111(tt, cons(y_0, n__take(y_1, y_2))) → U121(tt, cons(y_0, n__take(y_1, y_2)))
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, n__zeros)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U111(tt, cons(y_0, n__take(y_1, y_2))) → U121(tt, cons(y_0, n__take(y_1, y_2)))
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U121(tt, cons(y_0, n__take(y_1, y_2))) → LENGTH(cons(y_0, n__take(y_1, y_2)))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
U111(tt, n__take(y_0, y_1)) → U121(tt, n__take(y_0, y_1))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
LENGTH(cons(y0, x0)) → U111(tt, x0)
U111(tt, n__zeros) → U121(tt, n__zeros)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule LENGTH(cons(y0, x0)) → U111(tt, x0) we obtained the following new rules:
LENGTH(cons(x0, cons(y_0, y_1))) → U111(tt, cons(y_0, y_1))
LENGTH(cons(x0, cons(y_0, n__zeros))) → U111(tt, cons(y_0, n__zeros))
LENGTH(cons(x0, cons(y_0, n__take(y_1, y_2)))) → U111(tt, cons(y_0, n__take(y_1, y_2)))
LENGTH(cons(x0, n__zeros)) → U111(tt, n__zeros)
LENGTH(cons(x0, n__take(y_0, y_1))) → U111(tt, n__take(y_0, y_1))
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U111(tt, cons(y_0, n__take(y_1, y_2))) → U121(tt, cons(y_0, n__take(y_1, y_2)))
U121(tt, cons(y_0, n__take(y_1, y_2))) → LENGTH(cons(y_0, n__take(y_1, y_2)))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
LENGTH(cons(x0, cons(y_0, y_1))) → U111(tt, cons(y_0, y_1))
U111(tt, n__take(y_0, y_1)) → U121(tt, n__take(y_0, y_1))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
LENGTH(cons(x0, cons(y_0, n__zeros))) → U111(tt, cons(y_0, n__zeros))
LENGTH(cons(x0, cons(y_0, n__take(y_1, y_2)))) → U111(tt, cons(y_0, n__take(y_1, y_2)))
LENGTH(cons(x0, n__take(y_0, y_1))) → U111(tt, n__take(y_0, y_1))
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
U111(tt, n__zeros) → U121(tt, n__zeros)
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
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 dependency pairs:
LENGTH(cons(y0, n__take(x0, x1))) → U111(tt, take(activate(x0), activate(x1)))
LENGTH(cons(x0, cons(y_0, n__take(y_1, y_2)))) → U111(tt, cons(y_0, n__take(y_1, y_2)))
LENGTH(cons(x0, n__take(y_0, y_1))) → U111(tt, n__take(y_0, y_1))
Used ordering: POLO with Polynomial interpretation [25]:
POL(0) = 0
POL(LENGTH(x1)) = x1
POL(U111(x1, x2)) = x1 + x2
POL(U121(x1, x2)) = x1 + x2
POL(activate(x1)) = x1
POL(cons(x1, x2)) = 2·x1 + 2·x2
POL(n__take(x1, x2)) = 2 + 2·x1 + x2
POL(n__zeros) = 0
POL(take(x1, x2)) = 2 + 2·x1 + x2
POL(tt) = 0
POL(zeros) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U111(tt, cons(y_0, n__take(y_1, y_2))) → U121(tt, cons(y_0, n__take(y_1, y_2)))
U121(tt, cons(y_0, n__take(y_1, y_2))) → LENGTH(cons(y_0, n__take(y_1, y_2)))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
LENGTH(cons(x0, cons(y_0, y_1))) → U111(tt, cons(y_0, y_1))
U111(tt, n__take(y_0, y_1)) → U121(tt, n__take(y_0, y_1))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__take(x0, x1)) → U121(tt, take(activate(x0), activate(x1)))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
LENGTH(cons(x0, cons(y_0, n__zeros))) → U111(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, n__zeros)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(x0, cons(y_0, y_1))) → U111(tt, cons(y_0, y_1))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, cons(y_0, n__take(y_1, y_2))) → U121(tt, cons(y_0, n__take(y_1, y_2)))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
LENGTH(cons(x0, cons(y_0, n__zeros))) → U111(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, n__zeros)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(X1, X2) → n__take(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
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
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(x0, cons(y_0, y_1))) → U111(tt, cons(y_0, y_1))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, cons(y_0, n__take(y_1, y_2))) → U121(tt, cons(y_0, n__take(y_1, y_2)))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
LENGTH(cons(x0, cons(y_0, n__zeros))) → U111(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, n__zeros)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
R is empty.
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
zeros
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.
take(x0, x1)
activate(x0)
zeros
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(x0, cons(y_0, y_1))) → U111(tt, cons(y_0, y_1))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
U111(tt, cons(y_0, n__take(y_1, y_2))) → U121(tt, cons(y_0, n__take(y_1, y_2)))
LENGTH(cons(x0, cons(y_0, n__zeros))) → U111(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
U111(tt, n__zeros) → U121(tt, n__zeros)
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
R is empty.
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 dependency pairs:
U111(tt, cons(y_0, n__take(y_1, y_2))) → U121(tt, cons(y_0, n__take(y_1, y_2)))
Used ordering: POLO with Polynomial interpretation [25]:
POL(0) = 0
POL(LENGTH(x1)) = x1
POL(U111(x1, x2)) = 2·x1 + 2·x2
POL(U121(x1, x2)) = 2·x1 + x2
POL(cons(x1, x2)) = 2·x1 + 2·x2
POL(n__take(x1, x2)) = 2 + x1 + 2·x2
POL(n__zeros) = 0
POL(tt) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDPOrderProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(x0, cons(y_0, y_1))) → U111(tt, cons(y_0, y_1))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
LENGTH(cons(x0, cons(y_0, n__zeros))) → U111(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, n__zeros)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
R is empty.
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.
LENGTH(cons(x0, cons(y_0, y_1))) → U111(tt, cons(y_0, y_1))
LENGTH(cons(x0, cons(y_0, n__zeros))) → U111(tt, cons(y_0, n__zeros))
The remaining pairs can at least be oriented weakly.
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, n__zeros)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
| M( U121(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
| M( U111(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
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:
none
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
U111(tt, n__zeros) → U121(tt, n__zeros)
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule U111(tt, cons(y_0, y_1)) → U121(tt, cons(y_0, y_1)) we obtained the following new rules:
U111(tt, cons(0, n__zeros)) → U121(tt, cons(0, n__zeros))
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, cons(0, n__zeros)) → U121(tt, cons(0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, n__zeros)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule U111(tt, cons(y_0, n__zeros)) → U121(tt, cons(y_0, n__zeros)) we obtained the following new rules:
U111(tt, cons(0, n__zeros)) → U121(tt, cons(0, n__zeros))
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
U111(tt, cons(0, n__zeros)) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
U111(tt, n__zeros) → U121(tt, n__zeros)
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule U121(tt, cons(y_0, n__zeros)) → LENGTH(cons(y_0, n__zeros)) we obtained the following new rules:
U121(tt, cons(0, n__zeros)) → LENGTH(cons(0, n__zeros))
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, cons(0, n__zeros)) → LENGTH(cons(0, n__zeros))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, cons(0, n__zeros)) → U121(tt, cons(0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, n__zeros) → U121(tt, n__zeros)
U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1))
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule U121(tt, cons(y_0, y_1)) → LENGTH(cons(y_0, y_1)) we obtained the following new rules:
U121(tt, cons(0, n__zeros)) → LENGTH(cons(0, n__zeros))
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(0, n__zeros)) → LENGTH(cons(0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
U111(tt, cons(0, n__zeros)) → U121(tt, cons(0, n__zeros))
LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros)
U111(tt, n__zeros) → U121(tt, n__zeros)
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule LENGTH(cons(y0, n__zeros)) → U111(tt, n__zeros) we obtained the following new rules:
LENGTH(cons(0, n__zeros)) → U111(tt, n__zeros)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, cons(0, n__zeros)) → LENGTH(cons(0, n__zeros))
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, cons(0, n__zeros)) → U121(tt, cons(0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
LENGTH(cons(0, n__zeros)) → U111(tt, n__zeros)
U111(tt, n__zeros) → U121(tt, n__zeros)
LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule LENGTH(cons(y0, n__zeros)) → U111(tt, cons(0, n__zeros)) we obtained the following new rules:
LENGTH(cons(0, n__zeros)) → U111(tt, cons(0, n__zeros))
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ NonTerminationProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(0, n__zeros)) → LENGTH(cons(0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
U111(tt, cons(0, n__zeros)) → U121(tt, cons(0, n__zeros))
LENGTH(cons(0, n__zeros)) → U111(tt, n__zeros)
U111(tt, n__zeros) → U121(tt, n__zeros)
LENGTH(cons(0, n__zeros)) → U111(tt, cons(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 narrowing to the left:
The TRS P consists of the following rules:
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U121(tt, cons(0, n__zeros)) → LENGTH(cons(0, n__zeros))
U111(tt, n__zeros) → U121(tt, cons(0, n__zeros))
U111(tt, cons(0, n__zeros)) → U121(tt, cons(0, n__zeros))
LENGTH(cons(0, n__zeros)) → U111(tt, n__zeros)
U111(tt, n__zeros) → U121(tt, n__zeros)
LENGTH(cons(0, n__zeros)) → U111(tt, cons(0, n__zeros))
The TRS R consists of the following rules:none
s = LENGTH(cons(0, n__zeros)) evaluates to t =LENGTH(cons(0, n__zeros))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
LENGTH(cons(0, n__zeros)) → U111(tt, n__zeros)
with rule LENGTH(cons(0, n__zeros)) → U111(tt, n__zeros) at position [] and matcher [ ]
U111(tt, n__zeros) → U121(tt, n__zeros)
with rule U111(tt, n__zeros) → U121(tt, n__zeros) at position [] and matcher [ ]
U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
with rule U121(tt, n__zeros) → LENGTH(cons(0, n__zeros))
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
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
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
length(cons(x0, x1))
U21(tt, x0, x1, x2)
U12(tt, x0)
U22(tt, x0, x1, x2)
zeros
length(nil)
U23(tt, x0, x1, x2)
U11(tt, x0)
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.
length(cons(x0, x1))
U12(tt, x0)
length(nil)
U11(tt, x0)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
The TRS R consists of the following rules:
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
The set Q consists of the following terms:
take(x0, x1)
activate(x0)
U21(tt, x0, x1, x2)
U22(tt, x0, x1, x2)
zeros
U23(tt, x0, x1, x2)
We have to consider all minimal (P,Q,R)-chains.