Termination Proof using AProVE

Term Rewriting System R:
[X, XS, X1, X2]
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_TAIL}(cons(X, XS)) -> MARK(XS)
MARK(zeros) -> A_{_ZEROS}
MARK(tail(X)) -> A_{_TAIL}(mark(X))
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> A_{_TAIL}(mark(X))
A_{_TAIL}(cons(X, XS)) -> MARK(XS)

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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)
A_{_TAIL}(cons(X, XS)) -> MARK(XS)

The following usable rules for innermost can be oriented:

mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{tail, a_{_zeros}, mark, a_{_tail}} > cons
{tail, a_{_zeros}, mark, a_{_tail}} > zeros
{tail, a_{_zeros}, mark, a_{_tail}} > 0
{A_{_TAIL}, MARK}

resulting in one new DP problem.
Used Argument Filtering System:

A_{_TAIL}(x₁) -> A_{_TAIL}(x₁)
MARK(x₁) -> MARK(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
tail(x₁) -> tail(x₁)
mark(x₁) -> mark(x₁)
a_{_zeros} -> a_{_zeros}
a_{_tail}(x₁) -> a_{_tail}(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 2

             ↳Dependency Graph

Dependency Pair:

MARK(tail(X)) -> A_{_TAIL}(mark(X))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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