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

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳A-Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳ATrans
             ...
               →DP Problem 2
                 ↳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 w.r.t. the AFS can be oriented:

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


Used ordering: Lexicographic Path Order with Precedence:
trivial

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


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳ATrans
             ...
               →DP Problem 3
                 ↳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.

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