Term Rewriting System R:
[x, y, z, u]
+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)
+'(*(x, y), +(x, z)) -> +'(y, z)
+'(*(x, y), +(*(x, z), u)) -> +'(*(x, +(y, z)), u)
+'(*(x, y), +(*(x, z), u)) -> +'(y, z)

Furthermore, R contains one SCC.

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

Dependency Pairs:

+'(*(x, y), +(*(x, z), u)) -> +'(y, z)
+'(x, +(y, z)) -> +'(x, y)

Rules:

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Strategy:

innermost

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

+'(x, +(y, z)) -> +'(x, y)
two new Dependency Pairs are created:

+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(*(x'0, y''), +(+(*(x''', z''), u''), z)) -> +'(*(x'0, y''), +(*(x''', z''), u''))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

+'(*(x'0, y''), +(+(*(x''', z''), u''), z)) -> +'(*(x'0, y''), +(*(x''', z''), u''))
+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(*(x, y), +(*(x, z), u)) -> +'(y, z)

Rules:

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Strategy:

innermost

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

+'(*(x, y), +(*(x, z), u)) -> +'(y, z)
three new Dependency Pairs are created:

+'(*(x, *(x'0, y'')), +(*(x, +(*(x''', z''), u'')), u)) -> +'(*(x'0, y''), +(*(x''', z''), u''))
+'(*(x, y'), +(*(x, +(+(y'''', z''''), z'')), u)) -> +'(y', +(+(y'''', z''''), z''))
+'(*(x, *(x'0'', y'''')), +(*(x, +(+(*(x''''', z''''), u''''), z'')), u)) -> +'(*(x'0'', y''''), +(+(*(x''''', z''''), u''''), z''))

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 Pairs:

+'(*(x, *(x'0'', y'''')), +(*(x, +(+(*(x''''', z''''), u''''), z'')), u)) -> +'(*(x'0'', y''''), +(+(*(x''''', z''''), u''''), z''))
+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(*(x, y'), +(*(x, +(+(y'''', z''''), z'')), u)) -> +'(y', +(+(y'''', z''''), z''))
+'(*(x, *(x'0, y'')), +(*(x, +(*(x''', z''), u'')), u)) -> +'(*(x'0, y''), +(*(x''', z''), u''))
+'(*(x'0, y''), +(+(*(x''', z''), u''), z)) -> +'(*(x'0, y''), +(*(x''', z''), u''))

Rules:

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

+'(*(x, *(x'0'', y'''')), +(*(x, +(+(*(x''''', z''''), u''''), z'')), u)) -> +'(*(x'0'', y''''), +(+(*(x''''', z''''), u''''), z''))
+'(*(x, y'), +(*(x, +(+(y'''', z''''), z'')), u)) -> +'(y', +(+(y'''', z''''), z''))
+'(*(x, *(x'0, y'')), +(*(x, +(*(x''', z''), u'')), u)) -> +'(*(x'0, y''), +(*(x''', z''), u''))

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(*(x1, x2)) =  1 + x2 POL(+(x1, x2)) =  0 POL(+'(x1, x2)) =  x1

resulting in one new DP problem.

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

Dependency Pairs:

+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(*(x'0, y''), +(+(*(x''', z''), u''), z)) -> +'(*(x'0, y''), +(*(x''', z''), u''))

Rules:

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`

Dependency Pair:

+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))

Rules:

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Strategy:

innermost

The following dependency pair can be strictly oriented:

+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))

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

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(*(x1, x2)) =  0 POL(+(x1, x2)) =  1 + x1 + x2 POL(+'(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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