Term Rewriting System R:
[X, XS, N, X1, X2, Y, YS]
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
asel(0, cons(X, XS)) -> mark(X)
asel(s(N), cons(X, XS)) -> asel(mark(N), mark(XS))
asel(X1, X2) -> sel(X1, X2)
aminus(X, 0) -> 0
aminus(s(X), s(Y)) -> aminus(mark(X), mark(Y))
aminus(X1, X2) -> minus(X1, X2)
aquot(0, s(Y)) -> 0
aquot(s(X), s(Y)) -> s(aquot(aminus(mark(X), mark(Y)), s(mark(Y))))
aquot(X1, X2) -> quot(X1, X2)
azWquot(XS, nil) -> nil
azWquot(nil, XS) -> nil
azWquot(cons(X, XS), cons(Y, YS)) -> cons(aquot(mark(X), mark(Y)), zWquot(XS, YS))
azWquot(X1, X2) -> zWquot(X1, X2)
mark(from(X)) -> afrom(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(minus(X1, X2)) -> aminus(mark(X1), mark(X2))
mark(quot(X1, X2)) -> aquot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) -> azWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

AFROM(X) -> MARK(X)
ASEL(0, cons(X, XS)) -> MARK(X)
ASEL(s(N), cons(X, XS)) -> ASEL(mark(N), mark(XS))
ASEL(s(N), cons(X, XS)) -> MARK(N)
ASEL(s(N), cons(X, XS)) -> MARK(XS)
AMINUS(s(X), s(Y)) -> AMINUS(mark(X), mark(Y))
AMINUS(s(X), s(Y)) -> MARK(X)
AMINUS(s(X), s(Y)) -> MARK(Y)
AQUOT(s(X), s(Y)) -> AQUOT(aminus(mark(X), mark(Y)), s(mark(Y)))
AQUOT(s(X), s(Y)) -> AMINUS(mark(X), mark(Y))
AQUOT(s(X), s(Y)) -> MARK(X)
AQUOT(s(X), s(Y)) -> MARK(Y)
AZWQUOT(cons(X, XS), cons(Y, YS)) -> AQUOT(mark(X), mark(Y))
AZWQUOT(cons(X, XS), cons(Y, YS)) -> MARK(X)
AZWQUOT(cons(X, XS), cons(Y, YS)) -> MARK(Y)
MARK(from(X)) -> AFROM(mark(X))
MARK(from(X)) -> MARK(X)
MARK(sel(X1, X2)) -> ASEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(minus(X1, X2)) -> AMINUS(mark(X1), mark(X2))
MARK(minus(X1, X2)) -> MARK(X1)
MARK(minus(X1, X2)) -> MARK(X2)
MARK(quot(X1, X2)) -> AQUOT(mark(X1), mark(X2))
MARK(quot(X1, X2)) -> MARK(X1)
MARK(quot(X1, X2)) -> MARK(X2)
MARK(zWquot(X1, X2)) -> AZWQUOT(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) -> MARK(X1)
MARK(zWquot(X1, X2)) -> MARK(X2)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pairs:

ASEL(s(N), cons(X, XS)) -> MARK(XS)
ASEL(s(N), cons(X, XS)) -> MARK(N)
ASEL(s(N), cons(X, XS)) -> ASEL(mark(N), mark(XS))
AZWQUOT(cons(X, XS), cons(Y, YS)) -> MARK(Y)
AZWQUOT(cons(X, XS), cons(Y, YS)) -> MARK(X)
AQUOT(s(X), s(Y)) -> MARK(Y)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(zWquot(X1, X2)) -> MARK(X2)
MARK(zWquot(X1, X2)) -> MARK(X1)
AQUOT(s(X), s(Y)) -> MARK(X)
AZWQUOT(cons(X, XS), cons(Y, YS)) -> AQUOT(mark(X), mark(Y))
MARK(zWquot(X1, X2)) -> AZWQUOT(mark(X1), mark(X2))
MARK(quot(X1, X2)) -> MARK(X2)
MARK(quot(X1, X2)) -> MARK(X1)
AMINUS(s(X), s(Y)) -> MARK(Y)
AQUOT(s(X), s(Y)) -> AMINUS(mark(X), mark(Y))
AQUOT(s(X), s(Y)) -> AQUOT(aminus(mark(X), mark(Y)), s(mark(Y)))
MARK(quot(X1, X2)) -> AQUOT(mark(X1), mark(X2))
MARK(minus(X1, X2)) -> MARK(X2)
MARK(minus(X1, X2)) -> MARK(X1)
AMINUS(s(X), s(Y)) -> MARK(X)
AMINUS(s(X), s(Y)) -> AMINUS(mark(X), mark(Y))
MARK(minus(X1, X2)) -> AMINUS(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
ASEL(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> ASEL(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))
AFROM(X) -> MARK(X)

Rules:

afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
asel(0, cons(X, XS)) -> mark(X)
asel(s(N), cons(X, XS)) -> asel(mark(N), mark(XS))
asel(X1, X2) -> sel(X1, X2)
aminus(X, 0) -> 0
aminus(s(X), s(Y)) -> aminus(mark(X), mark(Y))
aminus(X1, X2) -> minus(X1, X2)
aquot(0, s(Y)) -> 0
aquot(s(X), s(Y)) -> s(aquot(aminus(mark(X), mark(Y)), s(mark(Y))))
aquot(X1, X2) -> quot(X1, X2)
azWquot(XS, nil) -> nil
azWquot(nil, XS) -> nil
azWquot(cons(X, XS), cons(Y, YS)) -> cons(aquot(mark(X), mark(Y)), zWquot(XS, YS))
azWquot(X1, X2) -> zWquot(X1, X2)
mark(from(X)) -> afrom(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(minus(X1, X2)) -> aminus(mark(X1), mark(X2))
mark(quot(X1, X2)) -> aquot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) -> azWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

AQUOT(s(X), s(Y)) -> AQUOT(aminus(mark(X), mark(Y)), s(mark(Y)))

Additionally, the following usable rules for innermost can be oriented:

mark(from(X)) -> afrom(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(minus(X1, X2)) -> aminus(mark(X1), mark(X2))
mark(quot(X1, X2)) -> aquot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) -> azWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil
azWquot(XS, nil) -> nil
azWquot(nil, XS) -> nil
azWquot(cons(X, XS), cons(Y, YS)) -> cons(aquot(mark(X), mark(Y)), zWquot(XS, YS))
azWquot(X1, X2) -> zWquot(X1, X2)
aquot(0, s(Y)) -> 0
aquot(s(X), s(Y)) -> s(aquot(aminus(mark(X), mark(Y)), s(mark(Y))))
aquot(X1, X2) -> quot(X1, X2)
aminus(X, 0) -> 0
aminus(s(X), s(Y)) -> aminus(mark(X), mark(Y))
aminus(X1, X2) -> minus(X1, X2)
asel(0, cons(X, XS)) -> mark(X)
asel(s(N), cons(X, XS)) -> asel(mark(N), mark(XS))
asel(X1, X2) -> sel(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(from(x1)) =  0 POL(MARK(x1)) =  1 POL(a__sel(x1, x2)) =  1 POL(a__zWquot(x1, x2)) =  0 POL(minus(x1, x2)) =  0 POL(A__FROM(x1)) =  1 POL(sel(x1, x2)) =  0 POL(mark(x1)) =  1 POL(a__from(x1)) =  0 POL(a__quot(x1, x2)) =  x2 POL(A__ZWQUOT(x1, x2)) =  1 POL(A__QUOT(x1, x2)) =  x1 POL(0) =  0 POL(cons(x1, x2)) =  0 POL(zWquot(x1, x2)) =  0 POL(a__minus(x1, x2)) =  0 POL(quot(x1, x2)) =  0 POL(nil) =  0 POL(s(x1)) =  1 POL(A__MINUS(x1, x2)) =  1 POL(A__SEL(x1, x2)) =  1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

ASEL(s(N), cons(X, XS)) -> MARK(XS)
ASEL(s(N), cons(X, XS)) -> MARK(N)
ASEL(s(N), cons(X, XS)) -> ASEL(mark(N), mark(XS))
AZWQUOT(cons(X, XS), cons(Y, YS)) -> MARK(Y)
AZWQUOT(cons(X, XS), cons(Y, YS)) -> MARK(X)
AQUOT(s(X), s(Y)) -> MARK(Y)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(zWquot(X1, X2)) -> MARK(X2)
MARK(zWquot(X1, X2)) -> MARK(X1)
AQUOT(s(X), s(Y)) -> MARK(X)
AZWQUOT(cons(X, XS), cons(Y, YS)) -> AQUOT(mark(X), mark(Y))
MARK(zWquot(X1, X2)) -> AZWQUOT(mark(X1), mark(X2))
MARK(quot(X1, X2)) -> MARK(X2)
MARK(quot(X1, X2)) -> MARK(X1)
AMINUS(s(X), s(Y)) -> MARK(Y)
AQUOT(s(X), s(Y)) -> AMINUS(mark(X), mark(Y))
MARK(quot(X1, X2)) -> AQUOT(mark(X1), mark(X2))
MARK(minus(X1, X2)) -> MARK(X2)
MARK(minus(X1, X2)) -> MARK(X1)
AMINUS(s(X), s(Y)) -> MARK(X)
AMINUS(s(X), s(Y)) -> AMINUS(mark(X), mark(Y))
MARK(minus(X1, X2)) -> AMINUS(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
ASEL(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> ASEL(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))
AFROM(X) -> MARK(X)

Rules:

afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
asel(0, cons(X, XS)) -> mark(X)
asel(s(N), cons(X, XS)) -> asel(mark(N), mark(XS))
asel(X1, X2) -> sel(X1, X2)
aminus(X, 0) -> 0
aminus(s(X), s(Y)) -> aminus(mark(X), mark(Y))
aminus(X1, X2) -> minus(X1, X2)
aquot(0, s(Y)) -> 0
aquot(s(X), s(Y)) -> s(aquot(aminus(mark(X), mark(Y)), s(mark(Y))))
aquot(X1, X2) -> quot(X1, X2)
azWquot(XS, nil) -> nil
azWquot(nil, XS) -> nil
azWquot(cons(X, XS), cons(Y, YS)) -> cons(aquot(mark(X), mark(Y)), zWquot(XS, YS))
azWquot(X1, X2) -> zWquot(X1, X2)
mark(from(X)) -> afrom(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(minus(X1, X2)) -> aminus(mark(X1), mark(X2))
mark(quot(X1, X2)) -> aquot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) -> azWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:36 minutes