Term Rewriting System R:
[x, y, z]
purge(nil) -> nil
purge(.(x, y)) -> .(x, purge(remove(x, y)))
remove(x, nil) -> nil
remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

PURGE(.(x, y)) -> PURGE(remove(x, y))
PURGE(.(x, y)) -> REMOVE(x, y)
REMOVE(x, .(y, z)) -> REMOVE(x, z)

Furthermore, R contains two SCCs.

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

Dependency Pair:

REMOVE(x, .(y, z)) -> REMOVE(x, z)

Rules:

purge(nil) -> nil
purge(.(x, y)) -> .(x, purge(remove(x, y)))
remove(x, nil) -> nil
remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))

The following dependency pair can be strictly oriented:

REMOVE(x, .(y, z)) -> REMOVE(x, z)

The following rules can be oriented:

purge(nil) -> nil
purge(.(x, y)) -> .(x, purge(remove(x, y)))
remove(x, nil) -> nil
remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
purge > . > {=, remove}

resulting in one new DP problem.
Used Argument Filtering System:
REMOVE(x1, x2) -> REMOVE(x1, x2)
.(x1, x2) -> .(x1, x2)
purge(x1) -> purge(x1)
remove(x1, x2) -> remove(x1, x2)
if(x1, x2, x3) -> x1
=(x1, x2) -> =(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pair:

Rules:

purge(nil) -> nil
purge(.(x, y)) -> .(x, purge(remove(x, y)))
remove(x, nil) -> nil
remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

PURGE(.(x, y)) -> PURGE(remove(x, y))

Rules:

purge(nil) -> nil
purge(.(x, y)) -> .(x, purge(remove(x, y)))
remove(x, nil) -> nil
remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))

The following dependency pair can be strictly oriented:

PURGE(.(x, y)) -> PURGE(remove(x, y))

The following rules can be oriented:

remove(x, nil) -> nil
remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))
purge(nil) -> nil
purge(.(x, y)) -> .(x, purge(remove(x, y)))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
purge > . > remove > =

resulting in one new DP problem.
Used Argument Filtering System:
PURGE(x1) -> PURGE(x1)
.(x1, x2) -> .(x1, x2)
remove(x1, x2) -> remove(x1, x2)
if(x1, x2, x3) -> x1
=(x1, x2) -> =(x1, x2)
purge(x1) -> purge(x1)

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

Dependency Pair:

Rules:

purge(nil) -> nil
purge(.(x, y)) -> .(x, purge(remove(x, y)))
remove(x, nil) -> nil
remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))

Using the Dependency Graph resulted in no new DP problems.

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