Term Rewriting System R:
[x, l, l1, l2]
isempty(nil) -> true
isempty(cons(x, l)) -> false
hd(cons(x, l)) -> x
tl(cons(x, l)) -> l
append(l1, l2) -> ifappend(l1, l2, l1)
ifappend(l1, l2, nil) -> l2
ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

APPEND(l1, l2) -> IFAPPEND(l1, l2, l1)
IFAPPEND(l1, l2, cons(x, l)) -> APPEND(l, l2)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pairs:

IFAPPEND(l1, l2, cons(x, l)) -> APPEND(l, l2)
APPEND(l1, l2) -> IFAPPEND(l1, l2, l1)

Rules:

isempty(nil) -> true
isempty(cons(x, l)) -> false
hd(cons(x, l)) -> x
tl(cons(x, l)) -> l
append(l1, l2) -> ifappend(l1, l2, l1)
ifappend(l1, l2, nil) -> l2
ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

The following dependency pair can be strictly oriented:

IFAPPEND(l1, l2, cons(x, l)) -> APPEND(l, l2)

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(IFAPPEND(x1, x2, x3)) =  x3 POL(cons(x1, x2)) =  1 + x2 POL(APPEND(x1, x2)) =  x1

resulting in one new DP problem.

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

Dependency Pair:

APPEND(l1, l2) -> IFAPPEND(l1, l2, l1)

Rules:

isempty(nil) -> true
isempty(cons(x, l)) -> false
hd(cons(x, l)) -> x
tl(cons(x, l)) -> l
append(l1, l2) -> ifappend(l1, l2, l1)
ifappend(l1, l2, nil) -> l2
ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

Using the Dependency Graph resulted in no new DP problems.

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