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

The following usable rules for innermost 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(MARK(x1)) =  x1 POL(0) =  0 POL(cons(x1, x2)) =  x1 + x2 POL(a__zeros) =  0 POL(tail(x1)) =  1 + x1 POL(a__tail(x1)) =  1 + x1 POL(mark(x1)) =  x1 POL(zeros) =  0 POL(A__TAIL(x1)) =  1 + x1

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)
azeros -> azeros
atail(x1) -> atail(x1)

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

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
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 the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 3`
`                 ↳Argument Filtering and Ordering`

Dependency Pair:

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

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 pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(cons(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pair:

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