Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(+(x₁, x₂)) = 1 + x₂

POL(+'(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 3

             ↳Dependency Graph

       →DP Problem 2

         ↳Nar

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

*'(*(x, y0), +(y'', z'')) -> *'(x, +(*(y0, y''), *(y0, z'')))
*'(*(x, *(x'', y'')), z'') -> *'(x, *(x'', *(y'', z'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Forward Instantiation Transformation

Dependency Pairs:

*'(*(x, *(x'', y'')), z'') -> *'(x, *(x'', *(y'', z'')))
*'(*(x, y0), +(y'', z'')) -> *'(x, +(*(y0, y''), *(y0, z'')))
*'(x, +(y, z)) -> *'(x, z)
*'(+(y, z), x) -> *'(x, z)
*'(+(y, z), x) -> *'(x, y)
*'(x, +(y, z)) -> *'(x, y)
*'(*(x, y), z) -> *'(y, z)

Rules:

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

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)

three new Dependency Pairs are created:

*'(+(y'', z''), +(y0, z)) -> *'(+(y'', z''), y0)
*'(*(x'', y''), +(y0, z)) -> *'(*(x'', y''), y0)
*'(*(x'', *(x'''', y'''')), +(y', z)) -> *'(*(x'', *(x'''', y'''')), y')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

*'(+(y'', z''), +(y0, z)) -> *'(+(y'', z''), y0)
*'(*(x'', *(x'''', y'''')), +(y', z)) -> *'(*(x'', *(x'''', y'''')), y')
*'(*(x'', y''), +(y0, z)) -> *'(*(x'', y''), y0)
*'(*(x, y0), +(y'', z'')) -> *'(x, +(*(y0, y''), *(y0, z'')))
*'(*(x, y), z) -> *'(y, z)
*'(+(y, z), x) -> *'(x, z)
*'(+(y, z), x) -> *'(x, y)
*'(x, +(y, z)) -> *'(x, z)
*'(*(x, *(x'', y'')), z'') -> *'(x, *(x'', *(y'', z'')))

Rules:

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

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, z)

eight new Dependency Pairs are created:

*'(x'', +(y, +(y'', z''))) -> *'(x'', +(y'', z''))
*'(+(y'', z'''), +(y, z'')) -> *'(+(y'', z'''), z'')
*'(*(x'', y''), +(y, z'')) -> *'(*(x'', y''), z'')
*'(*(x'', y0''), +(y, +(y'''', z''''))) -> *'(*(x'', y0''), +(y'''', z''''))
*'(*(x'', *(x'''', y'''')), +(y, z')) -> *'(*(x'', *(x'''', y'''')), z')
*'(+(y'''', z''''), +(y, +(y0'', z''))) -> *'(+(y'''', z''''), +(y0'', z''))
*'(*(x'''', y''''), +(y, +(y0'', z''))) -> *'(*(x'''', y''''), +(y0'', z''))
*'(*(x'''', *(x'''''', y'''''')), +(y, +(y''', z''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

*'(*(x'''', *(x'''''', y'''''')), +(y, +(y''', z''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))
*'(*(x'''', y''''), +(y, +(y0'', z''))) -> *'(*(x'''', y''''), +(y0'', z''))
*'(*(x'', *(x'''', y'''')), +(y, z')) -> *'(*(x'', *(x'''', y'''')), z')
*'(*(x'', y0''), +(y, +(y'''', z''''))) -> *'(*(x'', y0''), +(y'''', z''''))
*'(*(x'', y''), +(y, z'')) -> *'(*(x'', y''), z'')
*'(+(y'''', z''''), +(y, +(y0'', z''))) -> *'(+(y'''', z''''), +(y0'', z''))
*'(+(y'', z'''), +(y, z'')) -> *'(+(y'', z'''), z'')
*'(*(x'', *(x'''', y'''')), +(y', z)) -> *'(*(x'', *(x'''', y'''')), y')
*'(*(x'', y''), +(y0, z)) -> *'(*(x'', y''), y0)
*'(*(x, *(x'', y'')), z'') -> *'(x, *(x'', *(y'', z'')))
*'(x'', +(y, +(y'', z''))) -> *'(x'', +(y'', z''))
*'(*(x, y0), +(y'', z'')) -> *'(x, +(*(y0, y''), *(y0, z'')))
*'(*(x, y), z) -> *'(y, z)
*'(+(y, z), x) -> *'(x, z)
*'(+(y, z), x) -> *'(x, y)
*'(+(y'', z''), +(y0, z)) -> *'(+(y'', z''), y0)

Rules:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

*'(+(y0, z), +(y'', z'')) -> *'(+(y'', z''), y0)
*'(+(y0, z), *(x'', y'')) -> *'(*(x'', y''), y0)
*'(+(y', z), *(x'', *(x'''', y''''))) -> *'(*(x'', *(x'''', y'''')), y')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

*'(*(x'''', y''''), +(y, +(y0'', z''))) -> *'(*(x'''', y''''), +(y0'', z''))
*'(*(x'', *(x'''', y'''')), +(y, z')) -> *'(*(x'', *(x'''', y'''')), z')
*'(*(x'', y0''), +(y, +(y'''', z''''))) -> *'(*(x'', y0''), +(y'''', z''''))
*'(*(x'', y''), +(y, z'')) -> *'(*(x'', y''), z'')
*'(+(y0, z), +(y'', z'')) -> *'(+(y'', z''), y0)
*'(+(y'''', z''''), +(y, +(y0'', z''))) -> *'(+(y'''', z''''), +(y0'', z''))
*'(+(y'', z'''), +(y, z'')) -> *'(+(y'', z'''), z'')
*'(*(x'', *(x'''', y'''')), +(y', z)) -> *'(*(x'', *(x'''', y'''')), y')
*'(*(x'', y''), +(y0, z)) -> *'(*(x'', y''), y0)
*'(+(y', z), *(x'', *(x'''', y''''))) -> *'(*(x'', *(x'''', y'''')), y')
*'(+(y0, z), *(x'', y'')) -> *'(*(x'', y''), y0)
*'(+(y'', z''), +(y0, z)) -> *'(+(y'', z''), y0)
*'(*(x, *(x'', y'')), z'') -> *'(x, *(x'', *(y'', z'')))
*'(+(y, z), x) -> *'(x, z)
*'(x'', +(y, +(y'', z''))) -> *'(x'', +(y'', z''))
*'(*(x, y0), +(y'', z'')) -> *'(x, +(*(y0, y''), *(y0, z'')))
*'(*(x, y), z) -> *'(y, z)
*'(*(x'''', *(x'''''', y'''''')), +(y, +(y''', z''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))

Rules:

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

Strategy:

innermost

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

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

18 new Dependency Pairs are created:

*'(+(y, z''), +(y'', z''')) -> *'(+(y'', z'''), z'')
*'(+(y, z''), *(x'', y'')) -> *'(*(x'', y''), z'')
*'(+(y, +(y'''', z'''')), *(x'', y0'')) -> *'(*(x'', y0''), +(y'''', z''''))
*'(+(y, z'), *(x'', *(x'''', y''''))) -> *'(*(x'', *(x'''', y'''')), z')
*'(+(y, +(y0'', z'')), +(y'''', z'''')) -> *'(+(y'''', z''''), +(y0'', z''))
*'(+(y, +(y0'', z'')), *(x'''', y'''')) -> *'(*(x'''', y''''), +(y0'', z''))
*'(+(y, +(y''', z'')), *(x'''', *(x'''''', y''''''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))
*'(+(y, +(y'', +(y'''', z''''))), x') -> *'(x', +(y'', +(y'''', z'''')))
*'(+(y, +(y'', z''0')), +(y'''', z''''')) -> *'(+(y'''', z'''''), +(y'', z''0'))
*'(+(y, +(y'', z'''')), *(x'''', y'''')) -> *'(*(x'''', y''''), +(y'', z''''))
*'(+(y, +(y'', +(y'''''', z''''''))), *(x'''', y0'''')) -> *'(*(x'''', y0''''), +(y'', +(y'''''', z'''''')))
*'(+(y, +(y'', z''')), *(x'''', *(x'''''', y''''''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y'', z'''))
*'(+(y, +(y'', +(y0'''', z''''))), +(y'''''', z'''''')) -> *'(+(y'''''', z''''''), +(y'', +(y0'''', z'''')))
*'(+(y, +(y'', +(y0'''', z''''))), *(x'''''', y'''''')) -> *'(*(x'''''', y''''''), +(y'', +(y0'''', z'''')))
*'(+(y, +(y'', +(y''''', z''''))), *(x'''''', *(x'''''''', y''''''''))) -> *'(*(x'''''', *(x'''''''', y'''''''')), +(y'', +(y''''', z'''')))
*'(+(y, +(y'''', z'''')), +(y0'', z'')) -> *'(+(y0'', z''), +(y'''', z''''))
*'(+(y, *(x'''', y'''')), +(y0'', z'')) -> *'(+(y0'', z''), *(x'''', y''''))
*'(+(y, *(x'''', *(x'''''', y''''''))), +(y''', z'')) -> *'(+(y''', z''), *(x'''', *(x'''''', y'''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

Dependency Pairs:

*'(+(y, *(x'''', *(x'''''', y''''''))), +(y''', z'')) -> *'(+(y''', z''), *(x'''', *(x'''''', y'''''')))
*'(+(y, *(x'''', y'''')), +(y0'', z'')) -> *'(+(y0'', z''), *(x'''', y''''))
*'(+(y, +(y'''', z'''')), +(y0'', z'')) -> *'(+(y0'', z''), +(y'''', z''''))
*'(+(y, +(y'', +(y0'''', z''''))), +(y'''''', z'''''')) -> *'(+(y'''''', z''''''), +(y'', +(y0'''', z'''')))
*'(+(y, +(y'', z''0')), +(y'''', z''''')) -> *'(+(y'''', z'''''), +(y'', z''0'))
*'(+(y, +(y0'', z'')), +(y'''', z'''')) -> *'(+(y'''', z''''), +(y0'', z''))
*'(+(y, z''), +(y'', z''')) -> *'(+(y'', z'''), z'')
*'(+(y, +(y'', +(y''''', z''''))), *(x'''''', *(x'''''''', y''''''''))) -> *'(*(x'''''', *(x'''''''', y'''''''')), +(y'', +(y''''', z'''')))
*'(+(y, +(y'', +(y0'''', z''''))), *(x'''''', y'''''')) -> *'(*(x'''''', y''''''), +(y'', +(y0'''', z'''')))
*'(+(y, +(y'', z''')), *(x'''', *(x'''''', y''''''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y'', z'''))
*'(+(y, +(y'', +(y'''''', z''''''))), *(x'''', y0'''')) -> *'(*(x'''', y0''''), +(y'', +(y'''''', z'''''')))
*'(+(y, +(y'', z'''')), *(x'''', y'''')) -> *'(*(x'''', y''''), +(y'', z''''))
*'(+(y0, z), +(y'', z'')) -> *'(+(y'', z''), y0)
*'(+(y'''', z''''), +(y, +(y0'', z''))) -> *'(+(y'''', z''''), +(y0'', z''))
*'(+(y'', z'''), +(y, z'')) -> *'(+(y'', z'''), z'')
*'(+(y, +(y'', +(y'''', z''''))), x') -> *'(x', +(y'', +(y'''', z'''')))
*'(+(y, +(y''', z'')), *(x'''', *(x'''''', y''''''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))
*'(+(y, +(y0'', z'')), *(x'''', y'''')) -> *'(*(x'''', y''''), +(y0'', z''))
*'(+(y, z'), *(x'', *(x'''', y''''))) -> *'(*(x'', *(x'''', y'''')), z')
*'(+(y, +(y'''', z'''')), *(x'', y0'')) -> *'(*(x'', y0''), +(y'''', z''''))
*'(*(x'''', *(x'''''', y'''''')), +(y, +(y''', z''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))
*'(*(x'', *(x'''', y'''')), +(y, z')) -> *'(*(x'', *(x'''', y'''')), z')
*'(*(x'', y0''), +(y, +(y'''', z''''))) -> *'(*(x'', y0''), +(y'''', z''''))
*'(*(x'', y''), +(y, z'')) -> *'(*(x'', y''), z'')
*'(*(x'', *(x'''', y'''')), +(y', z)) -> *'(*(x'', *(x'''', y'''')), y')
*'(*(x'', y''), +(y0, z)) -> *'(*(x'', y''), y0)
*'(+(y, z''), *(x'', y'')) -> *'(*(x'', y''), z'')
*'(+(y', z), *(x'', *(x'''', y''''))) -> *'(*(x'', *(x'''', y'''')), y')
*'(+(y0, z), *(x'', y'')) -> *'(*(x'', y''), y0)
*'(+(y'', z''), +(y0, z)) -> *'(+(y'', z''), y0)
*'(*(x, *(x'', y'')), z'') -> *'(x, *(x'', *(y'', z'')))
*'(x'', +(y, +(y'', z''))) -> *'(x'', +(y'', z''))
*'(*(x, y0), +(y'', z'')) -> *'(x, +(*(y0, y''), *(y0, z'')))
*'(*(x, y), z) -> *'(y, z)
*'(*(x'''', y''''), +(y, +(y0'', z''))) -> *'(*(x'''', y''''), +(y0'', z''))

Rules:

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

Strategy:

innermost

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

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

11 new Dependency Pairs are created:

*'(*(x, *(x'', y'')), z'') -> *'(*(x'', y''), z'')
*'(*(x, *(x'', y0'')), +(y'''', z'''')) -> *'(*(x'', y0''), +(y'''', z''''))
*'(*(x, *(x'', *(x'''', y''''))), z') -> *'(*(x'', *(x'''', y'''')), z')
*'(*(x, *(x'''', y'''')), +(y0'', z'')) -> *'(*(x'''', y''''), +(y0'', z''))
*'(*(x, *(x'''', *(x'''''', y''''''))), +(y''', z'')) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))
*'(*(x, y0), +(y'', +(y'''', z''''))) -> *'(y0, +(y'', +(y'''', z'''')))
*'(*(x, *(x'''', y'''')), +(y'', z'''')) -> *'(*(x'''', y''''), +(y'', z''''))
*'(*(x, *(x'''', y0'''')), +(y'', +(y'''''', z''''''))) -> *'(*(x'''', y0''''), +(y'', +(y'''''', z'''''')))
*'(*(x, *(x'''', *(x'''''', y''''''))), +(y'', z''')) -> *'(*(x'''', *(x'''''', y'''''')), +(y'', z'''))
*'(*(x, *(x'''''', y'''''')), +(y'', +(y0'''', z''''))) -> *'(*(x'''''', y''''''), +(y'', +(y0'''', z'''')))
*'(*(x, *(x'''''', *(x'''''''', y''''''''))), +(y'', +(y''''', z''''))) -> *'(*(x'''''', *(x'''''''', y'''''''')), +(y'', +(y''''', z'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 9

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

*'(+(y, *(x'''', y'''')), +(y0'', z'')) -> *'(+(y0'', z''), *(x'''', y''''))
*'(+(y, +(y'''', z'''')), +(y0'', z'')) -> *'(+(y0'', z''), +(y'''', z''''))
*'(+(y, +(y'', +(y0'''', z''''))), +(y'''''', z'''''')) -> *'(+(y'''''', z''''''), +(y'', +(y0'''', z'''')))
*'(+(y, +(y'', z''0')), +(y'''', z''''')) -> *'(+(y'''', z'''''), +(y'', z''0'))
*'(+(y, +(y0'', z'')), +(y'''', z'''')) -> *'(+(y'''', z''''), +(y0'', z''))
*'(+(y, z''), +(y'', z''')) -> *'(+(y'', z'''), z'')
*'(+(y, +(y'', +(y''''', z''''))), *(x'''''', *(x'''''''', y''''''''))) -> *'(*(x'''''', *(x'''''''', y'''''''')), +(y'', +(y''''', z'''')))
*'(+(y, +(y'', +(y0'''', z''''))), *(x'''''', y'''''')) -> *'(*(x'''''', y''''''), +(y'', +(y0'''', z'''')))
*'(+(y, +(y'', z''')), *(x'''', *(x'''''', y''''''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y'', z'''))
*'(+(y, +(y'', +(y'''''', z''''''))), *(x'''', y0'''')) -> *'(*(x'''', y0''''), +(y'', +(y'''''', z'''''')))
*'(+(y, +(y'', z'''')), *(x'''', y'''')) -> *'(*(x'''', y''''), +(y'', z''''))
*'(+(y0, z), +(y'', z'')) -> *'(+(y'', z''), y0)
*'(+(y'''', z''''), +(y, +(y0'', z''))) -> *'(+(y'''', z''''), +(y0'', z''))
*'(+(y'', z'''), +(y, z'')) -> *'(+(y'', z'''), z'')
*'(+(y, +(y'', +(y'''', z''''))), x') -> *'(x', +(y'', +(y'''', z'''')))
*'(+(y, +(y''', z'')), *(x'''', *(x'''''', y''''''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))
*'(+(y, +(y0'', z'')), *(x'''', y'''')) -> *'(*(x'''', y''''), +(y0'', z''))
*'(+(y, z'), *(x'', *(x'''', y''''))) -> *'(*(x'', *(x'''', y'''')), z')
*'(+(y, +(y'''', z'''')), *(x'', y0'')) -> *'(*(x'', y0''), +(y'''', z''''))
*'(+(y, z''), *(x'', y'')) -> *'(*(x'', y''), z'')
*'(*(x, *(x'''''', *(x'''''''', y''''''''))), +(y'', +(y''''', z''''))) -> *'(*(x'''''', *(x'''''''', y'''''''')), +(y'', +(y''''', z'''')))
*'(*(x, *(x'''''', y'''''')), +(y'', +(y0'''', z''''))) -> *'(*(x'''''', y''''''), +(y'', +(y0'''', z'''')))
*'(*(x, *(x'''', *(x'''''', y''''''))), +(y'', z''')) -> *'(*(x'''', *(x'''''', y'''''')), +(y'', z'''))
*'(*(x, *(x'''', y0'''')), +(y'', +(y'''''', z''''''))) -> *'(*(x'''', y0''''), +(y'', +(y'''''', z'''''')))
*'(*(x, *(x'''', y'''')), +(y'', z'''')) -> *'(*(x'''', y''''), +(y'', z''''))
*'(*(x, y0), +(y'', +(y'''', z''''))) -> *'(y0, +(y'', +(y'''', z'''')))
*'(*(x, *(x'''', *(x'''''', y''''''))), +(y''', z'')) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))
*'(*(x, *(x'''', y'''')), +(y0'', z'')) -> *'(*(x'''', y''''), +(y0'', z''))
*'(*(x, *(x'', y0'')), +(y'''', z'''')) -> *'(*(x'', y0''), +(y'''', z''''))
*'(*(x'''', *(x'''''', y'''''')), +(y, +(y''', z''))) -> *'(*(x'''', *(x'''''', y'''''')), +(y''', z''))
*'(*(x'''', y''''), +(y, +(y0'', z''))) -> *'(*(x'''', y''''), +(y0'', z''))
*'(*(x'', *(x'''', y'''')), +(y, z')) -> *'(*(x'', *(x'''', y'''')), z')
*'(*(x'', y0''), +(y, +(y'''', z''''))) -> *'(*(x'', y0''), +(y'''', z''''))
*'(*(x'', y''), +(y, z'')) -> *'(*(x'', y''), z'')
*'(*(x'', *(x'''', y'''')), +(y', z)) -> *'(*(x'', *(x'''', y'''')), y')
*'(*(x, *(x'', *(x'''', y''''))), z') -> *'(*(x'', *(x'''', y'''')), z')
*'(*(x'', y''), +(y0, z)) -> *'(*(x'', y''), y0)
*'(*(x, *(x'', y'')), z'') -> *'(*(x'', y''), z'')
*'(+(y', z), *(x'', *(x'''', y''''))) -> *'(*(x'', *(x'''', y'''')), y')
*'(+(y'', z''), +(y0, z)) -> *'(+(y'', z''), y0)
*'(*(x, y0), +(y'', z'')) -> *'(x, +(*(y0, y''), *(y0, z'')))
*'(x'', +(y, +(y'', z''))) -> *'(x'', +(y'', z''))
*'(*(x, *(x'', y'')), z'') -> *'(x, *(x'', *(y'', z'')))
*'(+(y0, z), *(x'', y'')) -> *'(*(x'', y''), y0)
*'(+(y, *(x'''', *(x'''''', y''''''))), +(y''', z'')) -> *'(+(y''', z''), *(x'''', *(x'''''', y'''''')))

Rules:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:07 minutes

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