Term Rewriting System R:
[y, z, x]
f(cons(nil, y)) -> y
f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(cons(f(cons(nil, y)), z)) -> COPY(n, y, z)
COPY(0, y, z) -> F(z)
COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))
COPY(s(x), y, z) -> F(y)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pair:

COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))

Rules:

f(cons(nil, y)) -> y
f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

The following dependency pair can be strictly oriented:

COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))

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

f(cons(nil, y)) -> y
f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(COPY(x1, x2, x3)) =  x1 + x2 + x3 POL(s(x1)) =  1 + x1 POL(f(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
COPY(x1, x2, x3) -> COPY(x1, x2, x3)
s(x1) -> s(x1)
cons(x1, x2) -> x2
f(x1) -> f(x1)
copy(x1, x2, x3) -> x3

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

Dependency Pair:

Rules:

f(cons(nil, y)) -> y
f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

Using the Dependency Graph resulted in no new DP problems.

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