Term Rewriting System R:
[y, x, z]
or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(x, nil) -> false
mem(x, set(y)) -> =(x, y)
mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MEM(x, union(y, z)) -> OR(mem(x, y), mem(x, z))
MEM(x, union(y, z)) -> MEM(x, y)
MEM(x, union(y, z)) -> MEM(x, z)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`

Dependency Pairs:

MEM(x, union(y, z)) -> MEM(x, z)
MEM(x, union(y, z)) -> MEM(x, y)

Rules:

or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(x, nil) -> false
mem(x, set(y)) -> =(x, y)
mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

The following dependency pairs can be strictly oriented:

MEM(x, union(y, z)) -> MEM(x, z)
MEM(x, union(y, z)) -> MEM(x, y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MEM(x1, x2)) =  x2 POL(union(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

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

Dependency Pair:

Rules:

or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(x, nil) -> false
mem(x, set(y)) -> =(x, y)
mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

Using the Dependency Graph resulted in no new DP problems.

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