a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__sel(X1, X2) → sel(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(X1, X2) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__sel(X1, X2) → sel(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(X1, X2) → zWquot(X1, X2)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(quot(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → A__QUOT(a__minus(mark(X), mark(Y)), s(mark(Y)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
MARK(minus(X1, X2)) → A__MINUS(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → A__ZWQUOT(mark(X1), mark(X2))
MARK(minus(X1, X2)) → MARK(X2)
A__MINUS(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(Y)
A__QUOT(s(X), s(Y)) → MARK(Y)
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__MINUS(s(X), s(Y)) → MARK(Y)
MARK(quot(X1, X2)) → MARK(X1)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → A__QUOT(mark(X), mark(Y))
A__FROM(X) → MARK(X)
MARK(quot(X1, X2)) → A__QUOT(mark(X1), mark(X2))
A__MINUS(s(X), s(Y)) → MARK(X)
A__QUOT(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(X)
MARK(minus(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → MARK(X)
a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__sel(X1, X2) → sel(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(X1, X2) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(quot(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → A__QUOT(a__minus(mark(X), mark(Y)), s(mark(Y)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
MARK(minus(X1, X2)) → A__MINUS(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → A__ZWQUOT(mark(X1), mark(X2))
MARK(minus(X1, X2)) → MARK(X2)
A__MINUS(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(Y)
A__QUOT(s(X), s(Y)) → MARK(Y)
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__MINUS(s(X), s(Y)) → MARK(Y)
MARK(quot(X1, X2)) → MARK(X1)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → A__QUOT(mark(X), mark(Y))
A__FROM(X) → MARK(X)
MARK(quot(X1, X2)) → A__QUOT(mark(X1), mark(X2))
A__MINUS(s(X), s(Y)) → MARK(X)
A__QUOT(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(X)
MARK(minus(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → MARK(X)
a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__sel(X1, X2) → sel(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(X1, X2) → zWquot(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__QUOT(s(X), s(Y)) → A__QUOT(a__minus(mark(X), mark(Y)), s(mark(Y)))
Used ordering: Polynomial interpretation [25,35]:
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(quot(X1, X2)) → MARK(X2)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
MARK(minus(X1, X2)) → A__MINUS(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → A__ZWQUOT(mark(X1), mark(X2))
MARK(minus(X1, X2)) → MARK(X2)
A__MINUS(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(Y)
A__QUOT(s(X), s(Y)) → MARK(Y)
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__MINUS(s(X), s(Y)) → MARK(Y)
MARK(quot(X1, X2)) → MARK(X1)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → A__QUOT(mark(X), mark(Y))
A__FROM(X) → MARK(X)
MARK(quot(X1, X2)) → A__QUOT(mark(X1), mark(X2))
A__MINUS(s(X), s(Y)) → MARK(X)
A__QUOT(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(X)
MARK(minus(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → MARK(X)
The value of delta used in the strict ordering is 1/2.
POL(sel(x1, x2)) = 0
POL(from(x1)) = 0
POL(minus(x1, x2)) = 0
POL(A__ZWQUOT(x1, x2)) = 2
POL(A__MINUS(x1, x2)) = 2
POL(mark(x1)) = 1/4
POL(a__from(x1)) = (3/4)x_1
POL(a__minus(x1, x2)) = 0
POL(A__SEL(x1, x2)) = 2
POL(0) = 0
POL(a__zWquot(x1, x2)) = 1/4
POL(A__QUOT(x1, x2)) = 5/4 + (2)x_1 + x_2
POL(cons(x1, x2)) = 0
POL(MARK(x1)) = 2
POL(A__FROM(x1)) = 2
POL(a__sel(x1, x2)) = 1/4
POL(quot(x1, x2)) = x_2
POL(a__quot(x1, x2)) = x_2
POL(s(x1)) = 1/4
POL(zWquot(x1, x2)) = 0
POL(nil) = 1/4
a__minus(X, 0) → 0
a__from(X) → cons(mark(X), from(s(X)))
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
mark(from(X)) → a__from(mark(X))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
a__from(X) → from(X)
mark(nil) → nil
mark(0) → 0
mark(s(X)) → s(mark(X))
a__zWquot(X1, X2) → zWquot(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(minus(X1, X2)) → A__MINUS(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
MARK(quot(X1, X2)) → MARK(X2)
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → A__ZWQUOT(mark(X1), mark(X2))
MARK(minus(X1, X2)) → MARK(X2)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(Y)
A__MINUS(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__QUOT(s(X), s(Y)) → MARK(Y)
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__MINUS(s(X), s(Y)) → MARK(Y)
MARK(quot(X1, X2)) → MARK(X1)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → A__QUOT(mark(X), mark(Y))
A__FROM(X) → MARK(X)
MARK(quot(X1, X2)) → A__QUOT(mark(X1), mark(X2))
A__MINUS(s(X), s(Y)) → MARK(X)
A__QUOT(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(X)
MARK(minus(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → MARK(X)
a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__sel(X1, X2) → sel(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(X1, X2) → zWquot(X1, X2)