Term Rewriting System R:
[x, y, z]
app(app(., 1), x) -> x
app(app(., x), 1) -> x
app(app(., app(i, x)), x) -> 1
app(app(., x), app(i, x)) -> 1
app(app(., app(i, y)), app(app(., y), z)) -> z
app(app(., y), app(app(., app(i, y)), z)) -> z
app(app(., app(app(., x), y)), z) -> app(app(., x), app(app(., y), z))
app(i, 1) -> 1
app(i, app(i, x)) -> x
app(i, app(app(., x), y)) -> app(app(., app(i, y)), app(i, x))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(., app(app(., x), y)), z) -> APP(app(., x), app(app(., y), z))
APP(app(., app(app(., x), y)), z) -> APP(app(., y), z)
APP(app(., app(app(., x), y)), z) -> APP(., y)
APP(i, app(app(., x), y)) -> APP(app(., app(i, y)), app(i, x))
APP(i, app(app(., x), y)) -> APP(., app(i, y))
APP(i, app(app(., x), y)) -> APP(i, y)
APP(i, app(app(., x), y)) -> APP(i, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(i, app(app(., x), y)) -> APP(i, y)
APP(i, app(app(., x), y)) -> APP(app(., app(i, y)), app(i, x))
APP(app(., app(app(., x), y)), z) -> APP(app(., y), z)


Rules:


app(app(., 1), x) -> x
app(app(., x), 1) -> x
app(app(., app(i, x)), x) -> 1
app(app(., x), app(i, x)) -> 1
app(app(., app(i, y)), app(app(., y), z)) -> z
app(app(., y), app(app(., app(i, y)), z)) -> z
app(app(., app(app(., x), y)), z) -> app(app(., x), app(app(., y), z))
app(i, 1) -> 1
app(i, app(i, x)) -> x
app(i, app(app(., x), y)) -> app(app(., app(i, y)), app(i, x))


Strategy:

innermost




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

APP(i, app(app(., x), y)) -> APP(app(., app(i, y)), app(i, x))
three new Dependency Pairs are created:

APP(i, app(app(., x), app(i, x''))) -> APP(app(., x''), app(i, x))
APP(i, app(app(., x), app(app(., x''), y''))) -> APP(app(., app(app(., app(i, y'')), app(i, x''))), app(i, x))
APP(i, app(app(., app(i, x'')), y)) -> APP(app(., app(i, y)), x'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pairs:

APP(i, app(app(., app(i, x'')), y)) -> APP(app(., app(i, y)), x'')
APP(i, app(app(., x), app(app(., x''), y''))) -> APP(app(., app(app(., app(i, y'')), app(i, x''))), app(i, x))
APP(app(., app(app(., x), y)), z) -> APP(app(., y), z)
APP(i, app(app(., x), app(i, x''))) -> APP(app(., x''), app(i, x))
APP(i, app(app(., x), y)) -> APP(i, y)


Rules:


app(app(., 1), x) -> x
app(app(., x), 1) -> x
app(app(., app(i, x)), x) -> 1
app(app(., x), app(i, x)) -> 1
app(app(., app(i, y)), app(app(., y), z)) -> z
app(app(., y), app(app(., app(i, y)), z)) -> z
app(app(., app(app(., x), y)), z) -> app(app(., x), app(app(., y), z))
app(i, 1) -> 1
app(i, app(i, x)) -> x
app(i, app(app(., x), y)) -> app(app(., app(i, y)), app(i, x))


Strategy:

innermost




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

APP(i, app(app(., x), y)) -> APP(i, y)
four new Dependency Pairs are created:

APP(i, app(app(., x), app(app(., x''), y''))) -> APP(i, app(app(., x''), y''))
APP(i, app(app(., x), app(app(., x''), app(i, x'''')))) -> APP(i, app(app(., x''), app(i, x'''')))
APP(i, app(app(., x), app(app(., x''), app(app(., x''''), y'''')))) -> APP(i, app(app(., x''), app(app(., x''''), y'''')))
APP(i, app(app(., x), app(app(., app(i, x'''')), y''))) -> APP(i, app(app(., app(i, x'''')), y''))

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

APP(i, app(app(., x), app(app(., app(i, x'''')), y''))) -> APP(i, app(app(., app(i, x'''')), y''))
APP(i, app(app(., x), app(app(., x''), app(app(., x''''), y'''')))) -> APP(i, app(app(., x''), app(app(., x''''), y'''')))
APP(i, app(app(., x), app(app(., x''), app(i, x'''')))) -> APP(i, app(app(., x''), app(i, x'''')))
APP(i, app(app(., x), app(app(., x''), y''))) -> APP(i, app(app(., x''), y''))
APP(i, app(app(., x), app(app(., x''), y''))) -> APP(app(., app(app(., app(i, y'')), app(i, x''))), app(i, x))
APP(i, app(app(., x), app(i, x''))) -> APP(app(., x''), app(i, x))
APP(app(., app(app(., x), y)), z) -> APP(app(., y), z)
APP(i, app(app(., app(i, x'')), y)) -> APP(app(., app(i, y)), x'')


Rules:


app(app(., 1), x) -> x
app(app(., x), 1) -> x
app(app(., app(i, x)), x) -> 1
app(app(., x), app(i, x)) -> 1
app(app(., app(i, y)), app(app(., y), z)) -> z
app(app(., y), app(app(., app(i, y)), z)) -> z
app(app(., app(app(., x), y)), z) -> app(app(., x), app(app(., y), z))
app(i, 1) -> 1
app(i, app(i, x)) -> x
app(i, app(app(., x), y)) -> app(app(., app(i, y)), app(i, x))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes