Term Rewriting System R:
[f, x]
app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(F, app(app(F, f), x)), x) -> APP(app(F, app(G, app(app(F, f), x))), app(f, x))
APP(app(F, app(app(F, f), x)), x) -> APP(F, app(G, app(app(F, f), x)))
APP(app(F, app(app(F, f), x)), x) -> APP(G, app(app(F, f), x))
APP(app(F, app(app(F, f), x)), x) -> APP(f, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(F, app(app(F, f), x)), x) -> APP(f, x)
APP(app(F, app(app(F, f), x)), x) -> APP(app(F, app(G, app(app(F, f), x))), app(f, x))


Rule:


app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))


Strategy:

innermost




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

APP(app(F, app(app(F, f), x)), x) -> APP(app(F, app(G, app(app(F, f), x))), app(f, x))
one new Dependency Pair is created:

APP(app(F, app(app(F, app(F, app(app(F, f''), x''))), x'')), x'') -> APP(app(F, app(G, app(app(F, app(F, app(app(F, f''), x''))), x''))), app(app(F, app(G, app(app(F, f''), x''))), app(f'', 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(app(F, app(app(F, app(F, app(app(F, f''), x''))), x'')), x'') -> APP(app(F, app(G, app(app(F, app(F, app(app(F, f''), x''))), x''))), app(app(F, app(G, app(app(F, f''), x''))), app(f'', x'')))
APP(app(F, app(app(F, f), x)), x) -> APP(f, x)


Rule:


app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))


Strategy:

innermost




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

APP(app(F, app(app(F, f), x)), x) -> APP(f, x)
two new Dependency Pairs are created:

APP(app(F, app(app(F, app(F, app(app(F, f''), x'''))), x0)), x0) -> APP(app(F, app(app(F, f''), x''')), x0)
APP(app(F, app(app(F, app(F, app(app(F, app(F, app(app(F, f''''), x'''0))), x'''''))), x')), x') -> APP(app(F, app(app(F, app(F, app(app(F, f''''), x'''0))), x''''')), x')

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(app(F, app(app(F, app(F, app(app(F, app(F, app(app(F, f''''), x'''0))), x'''''))), x')), x') -> APP(app(F, app(app(F, app(F, app(app(F, f''''), x'''0))), x''''')), x')
APP(app(F, app(app(F, app(F, app(app(F, f''), x'''))), x0)), x0) -> APP(app(F, app(app(F, f''), x''')), x0)
APP(app(F, app(app(F, app(F, app(app(F, f''), x''))), x'')), x'') -> APP(app(F, app(G, app(app(F, app(F, app(app(F, f''), x''))), x''))), app(app(F, app(G, app(app(F, f''), x''))), app(f'', x'')))


Rule:


app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes