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

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

APP(app(:, z), app(app(+, x), app(f, y))) -> APP(app(:, app(app(g, z), y)), app(app(+, x), a))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

APP(app(:, z), app(app(+, x), app(f, y))) -> APP(app(g, z), y)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(:, app(app(+, x), y)), z) -> APP(app(:, x), z)
APP(app(:, app(app(+, x), y)), z) -> APP(app(+, app(app(:, x), z)), app(app(:, y), z))
APP(app(:, app(app(:, x), y)), z) -> APP(app(:, y), z)
APP(app(:, app(app(+, x), y)), z) -> APP(app(:, y), z)


Rules:


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


Strategy:

innermost




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

APP(app(:, app(app(+, x), y)), z) -> APP(app(+, app(app(:, x), z)), app(app(:, y), z))
six new Dependency Pairs are created:

APP(app(:, app(app(+, app(app(:, x''), y'')), y)), z'') -> APP(app(+, app(app(:, x''), app(app(:, y''), z''))), app(app(:, y), z''))
APP(app(:, app(app(+, app(app(+, x''), y'')), y)), z'') -> APP(app(+, app(app(+, app(app(:, x''), z'')), app(app(:, y''), z''))), app(app(:, y), z''))
APP(app(:, app(app(+, x0), y)), app(app(+, x''), app(f, y''))) -> APP(app(+, app(app(:, app(app(g, x0), y'')), app(app(+, x''), a))), app(app(:, y), app(app(+, x''), app(f, y''))))
APP(app(:, app(app(+, x), app(app(:, x''), y''))), z'') -> APP(app(+, app(app(:, x), z'')), app(app(:, x''), app(app(:, y''), z'')))
APP(app(:, app(app(+, x), app(app(+, x''), y''))), z'') -> APP(app(+, app(app(:, x), z'')), app(app(+, app(app(:, x''), z'')), app(app(:, y''), z'')))
APP(app(:, app(app(+, x), y0)), app(app(+, x''), app(f, y''))) -> APP(app(+, app(app(:, x), app(app(+, x''), app(f, y'')))), app(app(:, app(app(g, y0), y'')), app(app(+, x''), a)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(:, app(app(+, x), y0)), app(app(+, x''), app(f, y''))) -> APP(app(+, app(app(:, x), app(app(+, x''), app(f, y'')))), app(app(:, app(app(g, y0), y'')), app(app(+, x''), a)))
APP(app(:, app(app(+, x), app(app(+, x''), y''))), z'') -> APP(app(+, app(app(:, x), z'')), app(app(+, app(app(:, x''), z'')), app(app(:, y''), z'')))
APP(app(:, app(app(+, x), app(app(:, x''), y''))), z'') -> APP(app(+, app(app(:, x), z'')), app(app(:, x''), app(app(:, y''), z'')))
APP(app(:, app(app(+, x0), y)), app(app(+, x''), app(f, y''))) -> APP(app(+, app(app(:, app(app(g, x0), y'')), app(app(+, x''), a))), app(app(:, y), app(app(+, x''), app(f, y''))))
APP(app(:, app(app(+, app(app(+, x''), y'')), y)), z'') -> APP(app(+, app(app(+, app(app(:, x''), z'')), app(app(:, y''), z''))), app(app(:, y), z''))
APP(app(:, app(app(+, app(app(:, x''), y'')), y)), z'') -> APP(app(+, app(app(:, x''), app(app(:, y''), z''))), app(app(:, y), z''))
APP(app(:, app(app(+, x), y)), z) -> APP(app(:, y), z)
APP(app(:, app(app(:, x), y)), z) -> APP(app(:, y), z)
APP(app(:, app(app(+, x), y)), z) -> APP(app(:, x), z)


Rules:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:06 minutes