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)))

Innermost 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)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(REMOVE(x1, x2)) =  x1 + x2 POL(.(x1, x2)) =  1 + x1 + x2

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

Strategy:

innermost

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)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

The following usable rules for innermost w.r.t. to the AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(remove(x1, x2)) =  x1 + x2 POL(PURGE(x1)) =  x1 POL(=(x1, x2)) =  x1 + x2 POL(nil) =  0 POL(.(x1, x2)) =  1 + x1 + x2

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)

`   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)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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