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. 



   R
     ↳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. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)
       →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




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳Size-Change Principle
       →DP Problem 2
         ↳Nar


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. +'(+(x, y), z) -> +'(y, z)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
+(x1, x2) -> +(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →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: 



   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Polynomial Ordering


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




The following dependency pairs can be strictly oriented:

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


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

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


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Polo
             ...
               →DP Problem 5
                 ↳Dependency Graph


Dependency Pairs:

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




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


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Polo
             ...
               →DP Problem 6
                 ↳Usable Rules (Innermost)


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




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Polo
             ...
               →DP Problem 7
                 ↳Size-Change Principle


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. *'(*(x, y), z) -> *'(y, z)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
*(x1, x2) -> *(x1, x2)

We obtain no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:13 minutes