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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes