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

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, z)
:'(+(x, y), z) -> :'(y, z)
:'(z, +(x, f(y))) -> :'(g(z, y), +(x, a))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


:(:(x, y), z) -> :(x, :(y, z))
:(+(x, y), z) -> +(:(x, z), :(y, z))
:(z, +(x, f(y))) -> :(g(z, y), +(x, a))


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

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


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


Dependency Pairs:

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


Rules:


:(:(x, y), z) -> :(x, :(y, z))
:(+(x, y), z) -> +(:(x, z), :(y, z))
:(z, +(x, f(y))) -> :(g(z, y), +(x, a))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(:'(x1, x2))=  x1 + x2  
  POL(+(x1, x2))=  1 + x1 + x2  

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


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


Dependency Pair:


Rules:


:(:(x, y), z) -> :(x, :(y, z))
:(+(x, y), z) -> +(:(x, z), :(y, z))
:(z, +(x, f(y))) -> :(g(z, y), +(x, a))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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