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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

:'(:(:(:(C, x), y), z), u) -> :'(:(x, z), :(:(:(x, y), z), u))
:'(:(:(:(C, x), y), z), u) -> :'(x, z)
:'(:(:(:(C, x), y), z), u) -> :'(:(:(x, y), z), u)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, y), z)
:'(:(:(:(C, x), y), z), u) -> :'(x, y)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

:'(:(:(:(C, x), y), z), u) -> :'(x, y)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, y), z)
:'(:(:(:(C, x), y), z), u) -> :'(:(:(x, y), z), u)
:'(:(:(:(C, x), y), z), u) -> :'(x, z)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, z), :(:(:(x, y), z), u))


Rule:


:(:(:(:(C, x), y), z), u) -> :(:(x, z), :(:(:(x, y), z), u))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

:'(:(:(:(C, x), y), z), u) -> :'(x, y)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, y), z)
:'(:(:(:(C, x), y), z), u) -> :'(:(:(x, y), z), u)
:'(:(:(:(C, x), y), z), u) -> :'(x, z)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, z), :(:(:(x, y), z), u))


The following usable rule for innermost can be oriented:

:(:(:(:(C, x), y), z), u) -> :(:(x, z), :(:(:(x, y), z), u))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{:, :'}

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
             ↳Dependency Graph


Dependency Pair:


Rule:


:(:(:(:(C, x), y), z), u) -> :(:(x, z), :(:(:(x, y), z), u))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes