Term Rewriting System R:
[X, XS, X1, X2]
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

ATAIL(cons(X, XS)) -> MARK(XS)
MARK(zeros) -> AZEROS
MARK(tail(X)) -> ATAIL(mark(X))
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> ATAIL(mark(X))
ATAIL(cons(X, XS)) -> MARK(XS)

Rules:

azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0

The following dependency pair can be strictly oriented:

MARK(cons(X1, X2)) -> MARK(X1)

Additionally, the following usable rules using the Ce-refinement can be oriented:

mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
azeros -> cons(0, zeros)
azeros -> zeros

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(MARK(x1)) =  1 + x1 POL(a__zeros) =  1 POL(cons(x1, x2)) =  1 + x1 + x2 POL(tail(x1)) =  x1 POL(a__tail(x1)) =  x1 POL(mark(x1)) =  1 + x1 POL(zeros) =  0 POL(A__TAIL(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polynomial Ordering

Dependency Pairs:

MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> ATAIL(mark(X))
ATAIL(cons(X, XS)) -> MARK(XS)

Rules:

azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0

The following dependency pairs can be strictly oriented:

MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> ATAIL(mark(X))

Additionally, the following usable rules using the Ce-refinement can be oriented:

mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
azeros -> cons(0, zeros)
azeros -> zeros

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(MARK(x1)) =  x1 POL(a__zeros) =  0 POL(cons(x1, x2)) =  x2 POL(tail(x1)) =  1 + x1 POL(a__tail(x1)) =  1 + x1 POL(mark(x1)) =  x1 POL(zeros) =  0 POL(A__TAIL(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 3
Dependency Graph

Dependency Pair:

ATAIL(cons(X, XS)) -> MARK(XS)

Rules:

azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes