Term Rewriting System R:
[x, y, z]
+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))

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

Furthermore, R contains two SCCs. 


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


Dependency Pair:

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


Rules:


+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(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)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 3
             ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳FwdInst


Dependency Pair:

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


Rules:


+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
i > 0

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


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 3
             ↳AFS
             ...
               →DP Problem 4
                 ↳Dependency Graph
       →DP Problem 2
         ↳FwdInst


Dependency Pair:


Rules:


+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pairs:

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


Rules:


+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(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)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(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)
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(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)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 7
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(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)
six new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 8
                 ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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

+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
+ > 0

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


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 9
                 ↳Dependency Graph


Dependency Pair:


Rules:


+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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