Term Rewriting System R:
[x, y]
-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(-(x, y), -(x, y))
-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(x, y)

Furthermore, R contains one SCC.

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

Dependency Pairs:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(x, y)
-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(-(x, y), -(x, y))

Rule:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(x, y)

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

-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(-'(x1, x2)) =  1 + x1 + x2 POL(neg(x1)) =  x1 POL(-(x1, x2)) =  1 + x1 + x2

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

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

Dependency Pair:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(-(x, y), -(x, y))

Rule:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(-(x, y), -(x, y))

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

-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(-'(x1, x2)) =  1 + x1 + x2 POL(neg(x1)) =  1 + x1 POL(-(x1, x2)) =  1 + x1 + x2

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

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

Dependency Pair:

Rule:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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