Term Rewriting System R:
[h, g, x]
app(app(rec, h), app(g, 0)) -> g
app(app(rec, h), app(g, app(s, x))) -> app(app(h, x), app(app(rec, h), app(g, x)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(rec, h), app(g, app(s, x))) -> APP(app(h, x), app(app(rec, h), app(g, x)))
APP(app(rec, h), app(g, app(s, x))) -> APP(h, x)
APP(app(rec, h), app(g, app(s, x))) -> APP(app(rec, h), app(g, x))
APP(app(rec, h), app(g, app(s, x))) -> APP(g, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

APP(app(rec, h), app(g, app(s, x))) -> APP(g, x)
APP(app(rec, h), app(g, app(s, x))) -> APP(app(rec, h), app(g, x))
APP(app(rec, h), app(g, app(s, x))) -> APP(h, x)
APP(app(rec, h), app(g, app(s, x))) -> APP(app(h, x), app(app(rec, h), app(g, x)))


Rules:


app(app(rec, h), app(g, 0)) -> g
app(app(rec, h), app(g, app(s, x))) -> app(app(h, x), app(app(rec, h), app(g, x)))


Strategy:

innermost




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

APP(app(rec, h), app(g, app(s, x))) -> APP(h, x)
one new Dependency Pair is created:

APP(app(rec, app(rec, h'')), app(g, app(s, app(g'', app(s, x''))))) -> APP(app(rec, h''), app(g'', app(s, x'')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(rec, app(rec, h'')), app(g, app(s, app(g'', app(s, x''))))) -> APP(app(rec, h''), app(g'', app(s, x'')))
APP(app(rec, h), app(g, app(s, x))) -> APP(app(rec, h), app(g, x))
APP(app(rec, h), app(g, app(s, x))) -> APP(app(h, x), app(app(rec, h), app(g, x)))
APP(app(rec, h), app(g, app(s, x))) -> APP(g, x)


Rules:


app(app(rec, h), app(g, 0)) -> g
app(app(rec, h), app(g, app(s, x))) -> app(app(h, x), app(app(rec, h), app(g, x)))


Strategy:

innermost




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

APP(app(rec, h), app(g, app(s, x))) -> APP(g, x)
two new Dependency Pairs are created:

APP(app(rec, h), app(app(rec, h''), app(s, app(g'', app(s, x''))))) -> APP(app(rec, h''), app(g'', app(s, x'')))
APP(app(rec, h), app(app(rec, app(rec, h'''')), app(s, app(g'', app(s, app(g'''', app(s, x''''))))))) -> APP(app(rec, app(rec, h'''')), app(g'', app(s, app(g'''', app(s, x'''')))))

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

APP(app(rec, h), app(app(rec, app(rec, h'''')), app(s, app(g'', app(s, app(g'''', app(s, x''''))))))) -> APP(app(rec, app(rec, h'''')), app(g'', app(s, app(g'''', app(s, x'''')))))
APP(app(rec, h), app(app(rec, h''), app(s, app(g'', app(s, x''))))) -> APP(app(rec, h''), app(g'', app(s, x'')))
APP(app(rec, h), app(g, app(s, x))) -> APP(app(rec, h), app(g, x))
APP(app(rec, h), app(g, app(s, x))) -> APP(app(h, x), app(app(rec, h), app(g, x)))
APP(app(rec, app(rec, h'')), app(g, app(s, app(g'', app(s, x''))))) -> APP(app(rec, h''), app(g'', app(s, x'')))


Rules:


app(app(rec, h), app(g, 0)) -> g
app(app(rec, h), app(g, app(s, x))) -> app(app(h, x), app(app(rec, h), app(g, x)))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:03 minutes