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

Innermost 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
Argument Filtering and 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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


The following usable rules for innermost w.r.t. to the AFS can be oriented:

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
azeros -> cons(0, zeros)
azeros -> zeros


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{azeros, tail, mark, atail} > cons
{azeros, tail, mark, atail} > zeros
{azeros, tail, mark, atail} > 0
{ATAIL, MARK}

resulting in one new DP problem.
Used Argument Filtering System:
ATAIL(x1) -> ATAIL(x1)
MARK(x1) -> MARK(x1)
cons(x1, x2) -> cons(x1, x2)
tail(x1) -> tail(x1)
mark(x1) -> mark(x1)
atail(x1) -> atail(x1)
azeros -> azeros


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 2
Dependency Graph


Dependency Pair:

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


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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