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`
`         ↳Argument Filtering and 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)

The following rules can be oriented:

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))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MEM(x1, x2)) =  x1 + x2 POL(false) =  0 POL(union(x1, x2)) =  1 + x1 + x2 POL(or(x1, x2)) =  x1 + x2 POL(true) =  0 POL(nil) =  0 POL(set(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
MEM(x1, x2) -> MEM(x1, x2)
union(x1, x2) -> union(x1, x2)
or(x1, x2) -> or(x1, x2)
mem(x1, x2) -> x2
set(x1) -> set(x1)
=(x1, x2) -> x2

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →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