0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 QDP
↳7 MRRProof (⇔)
↳8 QDP
↳9 DependencyGraphProof (⇔)
↳10 AND
↳11 QDP
↳12 UsableRulesProof (⇔)
↳13 QDP
↳14 QDPSizeChangeProof (⇔)
↳15 TRUE
↳16 QDP
↳17 Narrowing (⇔)
↳18 QDP
↳19 DependencyGraphProof (⇔)
↳20 QDP
↳21 Narrowing (⇔)
↳22 QDP
↳23 DependencyGraphProof (⇔)
↳24 AND
↳25 QDP
↳26 UsableRulesProof (⇔)
↳27 QDP
↳28 Instantiation (⇔)
↳29 QDP
↳30 NonTerminationProof (⇔)
↳31 FALSE
↳32 QDP
a__zeros → cons(0, zeros)
a__and(tt, X) → mark(X)
a__length(nil) → 0
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(0, IL) → nil
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(a__and(x1, x2)) = x1 + x2
POL(a__length(x1)) = 1 + x1
POL(a__take(x1, x2)) = 1 + x1 + x2
POL(a__zeros) = 0
POL(and(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(length(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(tt) = 1
POL(zeros) = 0
a__and(tt, X) → mark(X)
a__length(nil) → 0
a__take(0, IL) → nil
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
A__LENGTH(cons(N, L)) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → MARK(L)
A__TAKE(s(M), cons(N, IL)) → MARK(N)
MARK(zeros) → A__ZEROS
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
A__LENGTH(cons(N, L)) → MARK(L)
MARK(and(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__LENGTH(cons(N, L)) → A__LENGTH(mark(L))
MARK(length(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → MARK(N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
A__LENGTH(cons(N, L)) → MARK(L)
MARK(and(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → MARK(N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
POL(0) = 0
POL(A__LENGTH(x1)) = 1 + x1
POL(A__TAKE(x1, x2)) = 1 + x1 + x2
POL(MARK(x1)) = x1
POL(a__and(x1, x2)) = 1 + x1 + x2
POL(a__length(x1)) = 2 + x1
POL(a__take(x1, x2)) = 2 + x1 + x2
POL(a__zeros) = 0
POL(and(x1, x2)) = 1 + x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(length(x1)) = 2 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 2 + x1 + x2
POL(tt) = 0
POL(zeros) = 0
A__LENGTH(cons(N, L)) → A__LENGTH(mark(L))
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
From the DPs we obtained the following set of size-change graphs:
A__LENGTH(cons(N, L)) → A__LENGTH(mark(L))
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
A__LENGTH(cons(y0, zeros)) → A__LENGTH(a__zeros)
A__LENGTH(cons(y0, and(x0, x1))) → A__LENGTH(a__and(mark(x0), x1))
A__LENGTH(cons(y0, length(x0))) → A__LENGTH(a__length(mark(x0)))
A__LENGTH(cons(y0, take(x0, x1))) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(y0, cons(x0, x1))) → A__LENGTH(cons(mark(x0), x1))
A__LENGTH(cons(y0, 0)) → A__LENGTH(0)
A__LENGTH(cons(y0, tt)) → A__LENGTH(tt)
A__LENGTH(cons(y0, nil)) → A__LENGTH(nil)
A__LENGTH(cons(y0, s(x0))) → A__LENGTH(s(mark(x0)))
A__LENGTH(cons(y0, zeros)) → A__LENGTH(a__zeros)
A__LENGTH(cons(y0, and(x0, x1))) → A__LENGTH(a__and(mark(x0), x1))
A__LENGTH(cons(y0, length(x0))) → A__LENGTH(a__length(mark(x0)))
A__LENGTH(cons(y0, take(x0, x1))) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(y0, cons(x0, x1))) → A__LENGTH(cons(mark(x0), x1))
A__LENGTH(cons(y0, 0)) → A__LENGTH(0)
A__LENGTH(cons(y0, tt)) → A__LENGTH(tt)
A__LENGTH(cons(y0, nil)) → A__LENGTH(nil)
A__LENGTH(cons(y0, s(x0))) → A__LENGTH(s(mark(x0)))
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
A__LENGTH(cons(y0, zeros)) → A__LENGTH(a__zeros)
A__LENGTH(cons(y0, and(x0, x1))) → A__LENGTH(a__and(mark(x0), x1))
A__LENGTH(cons(y0, length(x0))) → A__LENGTH(a__length(mark(x0)))
A__LENGTH(cons(y0, take(x0, x1))) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(y0, cons(x0, x1))) → A__LENGTH(cons(mark(x0), x1))
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
A__LENGTH(cons(y0, zeros)) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(y0, zeros)) → A__LENGTH(zeros)
A__LENGTH(cons(y0, and(x0, x1))) → A__LENGTH(a__and(mark(x0), x1))
A__LENGTH(cons(y0, length(x0))) → A__LENGTH(a__length(mark(x0)))
A__LENGTH(cons(y0, take(x0, x1))) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(y0, cons(x0, x1))) → A__LENGTH(cons(mark(x0), x1))
A__LENGTH(cons(y0, zeros)) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(y0, zeros)) → A__LENGTH(zeros)
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
A__LENGTH(cons(y0, zeros)) → A__LENGTH(cons(0, zeros))
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
A__LENGTH(cons(y0, zeros)) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(0, zeros)) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(0, zeros)) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(y0, and(x0, x1))) → A__LENGTH(a__and(mark(x0), x1))
A__LENGTH(cons(y0, length(x0))) → A__LENGTH(a__length(mark(x0)))
A__LENGTH(cons(y0, take(x0, x1))) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(y0, cons(x0, x1))) → A__LENGTH(cons(mark(x0), x1))
a__zeros → cons(0, zeros)
a__length(cons(N, L)) → s(a__length(mark(L)))
a__take(s(M), cons(N, IL)) → cons(mark(N), take(M, IL))
mark(zeros) → a__zeros
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
mark(s(X)) → s(mark(X))
a__zeros → zeros
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)