Term Rewriting System R:
[x, y]
merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MERGE(++(x, y), ++(u, v)) -> MERGE(y, ++(u, v))
MERGE(++(x, y), ++(u, v)) -> MERGE(++(x, y), v)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`

Dependency Pair:

MERGE(++(x, y), ++(u, v)) -> MERGE(y, ++(u, v))

Rules:

merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

MERGE(++(x, y), ++(u, v)) -> MERGE(y, ++(u, v))
one new Dependency Pair is created:

MERGE(++(x, ++(x'', y'')), ++(u, v)) -> MERGE(++(x'', y''), ++(u, v))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Forward Instantiation Transformation`

Dependency Pair:

MERGE(++(x, ++(x'', y'')), ++(u, v)) -> MERGE(++(x'', y''), ++(u, v))

Rules:

merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

MERGE(++(x, ++(x'', y'')), ++(u, v)) -> MERGE(++(x'', y''), ++(u, v))
one new Dependency Pair is created:

MERGE(++(x, ++(x'''', ++(x''''', y''''))), ++(u, v)) -> MERGE(++(x'''', ++(x''''', y'''')), ++(u, v))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pair:

MERGE(++(x, ++(x'''', ++(x''''', y''''))), ++(u, v)) -> MERGE(++(x'''', ++(x''''', y'''')), ++(u, v))

Rules:

merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

Strategy:

innermost

The following dependency pair can be strictly oriented:

MERGE(++(x, ++(x'''', ++(x''''', y''''))), ++(u, v)) -> MERGE(++(x'''', ++(x''''', y'''')), ++(u, v))

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(v) =  0 POL(++(x1, x2)) =  1 + x2 POL(MERGE(x1, x2)) =  1 + x1 POL(u) =  0

resulting in one new DP problem.

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

Dependency Pair:

Rules:

merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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